Vincent Jomelli
email: jomelli@cerege.fr

Patrick Wagnon

Didier Swingedouw

Joanna Charton

Régis Braucher

Adèle Hue

Fanny Brun

Christophe Colin

Stephanie Gairoard

Dibas Shrestha

Aster Team

Georges Aumaître

Karim Keddadouche

Fawzi Zaidi

Unraveling the climate control on debris-free glacier evolution in the Everest region (Nepal, central Himalaya) during the Holocene

Keywords: Nepalese glacier, 10Be chronology, Holocene, AMOC, Indian monsoon

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

these observations on mass balance variability only explore the recent decades, studies on paleo glacial extents are useful to investigate the long-term climate forcing on glacier evolution. To do so, we investigated the long-term evolution of the debris-free Mera glacier and a neighbouring small debris-free South Khare glacier. Fifty-one 10 Be CRE ages were obtained from samples collected on moraine boulders and roches moutonnées. 10 Be CRE ages of the boulders span from the end of the Lateglacial (19.0 -11.7 ka) to the Little Ice Age (~ 0.6-0.1 ka). The oldest dated moraine in this study was observed at the base of South Khare glacier with an age of 13.6 ± 0.5 ka. The two glaciers subsequently experienced their largest Holocene extent in the Early Holocene with moraines dated to 11.0 ± 0.3 ka at the base of Mera glacier and 10.8 ± 0.5 ka at the base of South Khare glacier. We did not observe any moraine from the Mid-Holocene. During the Late Holocene several glacier advances were recorded around 2.3 ± 0.2 ka, 1.5 ka and then during the last centuries at Mera glacier and around 2.8 ± 0.6 ka, and during the Little Ice Age at South Khare glacier. To explore the links between long-term Nepalese glacier changes and climate, we used oceanic and terrestrial Indian Summer monsoon reconstructions and temperature and precipitation output from two transient global climate models TraCE and LOVECLIM. These climate data outputs were corrected by a reconstruction of the Atlantic Meridional Overturning Circulation (AMOC) over the Holocene and its associated climatic impacts. We also used sensitivity experiments from the IPSL (Institut Pierre Simon Laplace) model to discuss the possible influence of horizontal resolution, land hydrology, vegetation and runoff on changes in Asian summer monsoon. Importantly, we show this long-term Nepalese glacier pattern does not perfectly conform neither to the Indian monsoon precipitation that is documented from terrestrial and marine records nor to temperature and precipitation changes simulated by the models. While the maximum glacier extent in the Early Holocene corresponds to enhanced precipitation documented by proxies and models, the Late Holocene glacier advance remains puzzling. We claim that new paleo glacier records and improved climate simulations are necessary to get a better understanding of past glacier changes and the associated climate dynamics, which might be crucial to gain confidence in both glacier and climate future evolutions.

1-Introduction

Over the last decades the recent evolution of the High Mountain Asian (HMA) glaciers mass balance has been widely investigated as they contribute importantly to water resources for approximately 800 million people and are a component of global sea level rise [START_REF] Brun | A spatially resolved estimate of high mountain Asia glacier mass balances from 2000 to 2016[END_REF][START_REF] Pritchard | Asia's shrinking glaciers protect large populations from drought stress[END_REF][START_REF] Azam | Glaciohydrology of the Himalaya-Karakoram[END_REF]. Recent studies covering most HMA glaciers have revealed an overall mass loss over the last decades with values about -0.18 ± 0.04 m of water equivalent per year (w.e. yr-1) with strong spatial and individual glacier mass balance variability [START_REF] Brun | A spatially resolved estimate of high mountain Asia glacier mass balances from 2000 to 2016[END_REF][START_REF] Hugonnet | Accelerated global glacier mass loss in the early twenty-first century[END_REF]. This large spatio-temporal mass balance variability is mainly due to climate and geomorphological causes. Along the HMA temperature and precipitation vary spatially and temporally, leading to heterogeneous climate change and behavior of glaciers (e.g., [START_REF] Sakaï | Contrasting glacier responses to recent climate change in high mountain Asia[END_REF]Yan et al., 2018[START_REF] Yan | Deciphering the evolution and forcing mechanisms of glaciation over the Himalayan-Tibetan orogen during the past 20,000 years[END_REF][START_REF] Azam | Glaciohydrology of the Himalaya-Karakoram[END_REF]. In addition, [START_REF] Sakaï | Contrasting glacier responses to recent climate change in high mountain Asia[END_REF] evidenced a spatially variable sensitivity of mass balances of HMA glaciers to temperature while their behavior also depends on precipitation changes.

The other main cause of mass balance variability is related to the geomorphological context of individual glaciers. [START_REF] Brun | Heterogeneous influence of glacier morphology on the mass balance variability in high mountain Asia[END_REF] have reported that the slope of the glacier tongue, the mean glacier elevation, the percentage of supraglacial debris cover, the avalanche contributing area and the presence of a lake at the base of glaciers all together explain between 11 and 48% of the current mass balance variability. However, as these contributions only explore the recent decades, studies on paleo glacial extents during the Holocene and older periods are needed to investigate potential influence of long-term climate forcing on multi-centennial to millennial glacier evolution and thus offer an opportunity to put the recent glacier decline in a longer perspective.

Previous studies that investigated glacier fluctuations during the Lateglacial and the Holocene (11.7 ka -present) using in situ produced 10 Be cosmic ray exposure (CRE) ages have shown that the multi-centennial glacier evolution differs across the HMA. This is assumed to reflect the spatial variability of precipitation and temperature as currently observed. This is related to relative influence of the mid-latitude westerlies, the Indian Summer Monsoon (ISM) and the East Asian Summer Monsoon (ESM) [START_REF] Owen | Quaternary glaciations of the Himalayan-Tibetan orogen[END_REF][START_REF] Owen | Quaternary glaciation of Mount Everest[END_REF][START_REF] Owen | Latest pleistocene and holocene glacier fluctuations in the Himalaya and Tibet[END_REF][START_REF] Owen | Nature and timing of Quaternary glaciation in the Himalayan-Tibetan orogen[END_REF]Saha et al., 2019). While the mid-latitude westerlies might significantly influence the western end of the range and Tibet [START_REF] Mölg | Mid-latitude westerlies as a driver of glacier variability in monsoonal High Asia[END_REF], the ISM and ESM both combined in the Asian Summer Monsoon are believed to largely dominate the southern and eastern regions, and to a lesser extent the interior (continental) portions of the HMA.

As it is observed that the geomorphological context has influence on mass balance variability over the current period, this geomorphological context should also be considered while investigating long-term glacier change. For instance, in the Khumbu valley, (Everest region, Nepal) previous studies of glacier fluctuations from the Lateglacial were conducted on large debris-covered glaciers [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF][START_REF] Hornsey | Be-10 dating of ice-marginal moraines in the Khumbu Valley, Nepal, Central Himalaya, reveals the response of monsoon-influenced debris-covered glaciers to Holocene climate change[END_REF]. However, a detailed understanding of the Lateglacial-Holocene glacier history from debris-free glaciers in the central Himalaya is lacking (Saha et al., 2019). Thus it remains still unclear if the debris cover in the ablation zone may have influenced the long-term glacier response to climate change compared to debris-free glaciers [START_REF] Owen | Climatic and topographic controls on the style and timing of Late Quaternary glaciation throughout Tibet and the Himalaya defined by 10Be cosmogenic radionuclide surface exposure dating[END_REF].

In this context we investigated the fluctuations of two debris-free glaciers in Nepal (Everest region (Fig. 1)), based on 51 10 Be CRE ages determined from rock samples of moraine boulders (n=43) and roche moutonnées (n=8) whose ages span the Lateglacial and the Holocene. The main goals of this study are to document major periods of advances of these two glaciers at a multi-centennial time scale during the Holocene, to compare this signal with other glaciers in Nepal and investigate the potential long-term climate forcings responsible for their evolution.

Among the two investigated glaciers, the Mera glacier is an ideal target to explore long-term glacier-climate relationship and compare long-term vs. short-term glacier response to climate change. Indeed, on this glacier an understanding of physical processes driving current mass balance has been led through mass and energy balances surveys since 2007 [START_REF] Wagnon | Seasonal and annual mass balances of Mera and Pokalde glaciers (Nepal Himalaya) since 2007[END_REF]2021;[START_REF] Sherpa | Contrasted surface mass balances of debris-free glaciers observed between the southern and the inner parts of the Everest region[END_REF][START_REF] Litt | Glacier ablation and temperature indexed melt models in the nepalese himalaya[END_REF]. The originality of this study is thus to combine different approaches i.e., surface process understanding at present, as well as moraine dating, climate reconstructions and climate model outputs over the Lateglacial-Holocene period, to unravel the climate control on debris-free glacier evolution in the Everest region. The study area, moraine stratigraphy and the climate setting are presented in section 2. Based on the methodology described in section 3, CRE moraine ages are presented in section 4. In section 5-1, 5-2 and 5-3 we discuss our moraine records and compare the two chronologies with other studies conducted in Nepal and in other regions of HMA respectively. In section 5-4 we discuss potential forcings driving the evolution of the two investigated glaciers.

2-Study Area and climate setting

2-1 Physical setting

The two investigated Mera (ME) and South Khare (SK) glaciers are located in the upper Dudh Koshi basin, at about 30 km south of Everest summit (central Himalaya, 27°.74 Lat; 86°.86 Long) (Fig. 1). In this area the bedrock is dominated by quartz-rich granite and gneiss. Mera glacier is a debris-free summer-accumulation glacier type with an area of 4.84 ± 0.34 km² in 2018 and a maximum altitude of 6420 m a.s.l. [START_REF] Wagnon | Reanalysing the 2007-19 glaciological massbalance series of mera glacier, Nepal, central Himalaya, using geodetic mass balance[END_REF]. The glacier extends northwards and divides into two branches at 5800 m a.s.l.. The main branch, oriented north-west, with a mean slope of ~ 16°, goes down to 4910 m a.s.l. at snout. Here, samples from several moraines were collected. The minor Naulek branch, oriented north-east was not investigated in this study. Mera glacier mass balance has been investigated both using the classical glaciological (stakes) method since 2007 and more recently the geodetic method [START_REF] Wagnon | Seasonal and annual mass balances of Mera and Pokalde glaciers (Nepal Himalaya) since 2007[END_REF]2021). Annual and seasonal measurements of the glacier mass balance showed that the accumulation and ablation processes happen during the same monsoon period.

There is limited mass change during the dry, cold and windy winter, apart from wind ablation and sublimation [START_REF] Sherpa | Contrasted surface mass balances of debris-free glaciers observed between the southern and the inner parts of the Everest region[END_REF][START_REF] Litt | Glacier ablation and temperature indexed melt models in the nepalese himalaya[END_REF][START_REF] Wagnon | Reanalysing the 2007-19 glaciological massbalance series of mera glacier, Nepal, central Himalaya, using geodetic mass balance[END_REF]. The glacier-wide mass balance calibrated over geodetic mass balance was slightly negative for the period 2007-2013 (MB -0.23 +/-0.23 m w.e./yr) with more negative values for the period 2013-2020 (-0.57 +/-0.19 m w.e./yr) [START_REF] Wagnon | Reanalysing the 2007-19 glaciological massbalance series of mera glacier, Nepal, central Himalaya, using geodetic mass balance[END_REF]. In addition the mass balance variations of this glacier are in agreement with observations conducted at a regional scale [START_REF] Hugonnet | Accelerated global glacier mass loss in the early twenty-first century[END_REF].Thus, it can be seen as representative of the regional trend. Mera glacier is bounded by six distinct lateral moraines, which were named, from the right to the left side of the glacier looking downhill, M1 for the youngest to Mx for the oldest i. This lateral moraine shows two distinct crests and only the distal one has been investigated. This right lateral ME-M4 moraine has an asymmetrical cross-profile with a rather flat surface and a steep but stable, vegetated distal slope (Figs 2,3). Downslope the moraine ME-M4 was probably connected to the lateral moraine of South Khare glacier. The crest is rounded and has a relief of about 20 m above the valley floor. Moraines on the left side of the glacier are less well preserved. Only two ridges were investigated. One is relatively fresh in shape (ME-M5) while the other (ME-M6) is also covered by vegetation. From the terminus of the frontal moraine a fragmented ridge is only identifiable over about 100 m toward the glacier. At an altitude of about 4750 m a.s.l. the ridge of ME-M5 is buried by another moraine which was not studied. This part of the left side of the glacier was also impacted by dirty snow avalanches and was not deeper investigated. The relief of this left lateral moraine ME-M5 is about 50 m above the valley. ME-M6 moraine is only preserved from periglacial processes over a distance of approximately 100 m. This vegetated moraine constitutes the most external ridge of the rampart on the left side of the glacier. On the inner side of these two moraines, several ridges are visible but are too steep to be sampled. South Khare glacier (0.47 km 2 ) is located 1 km north of Mera glacier and has a catchment area of ~2.7 km² with an elevation ranging from ~5200 m a.s.l at the front to ~5890 m a.s.l at the top (Fig. 1). It is also a debris-free glacier facing west with a mean slope of ~ 17°. At the base of South Khare glacier, the configuration of the moraines is rather similar to that of Mera glacier with eight identified moraines the SK M1-8 moraines, with each crest separated by 2-70 m horizontally (Figs. 4,5). At a distance of ~700 m down-valley from the current front position, a large rampart of about 60 m high is composed of at least two young nested lateral moraines with individual crests (SK-M1, -M2 on the right side, -M6, -M7 on the left side) that were sampled at both sides of the valley. The South Khare right lateral moraine SK-M1 is continuous along the right of the glacier with an elevation ranging from 5010 to 5200 m a.s.l..

2-2 Climate Setting

The Everest region is under the influence of the ISM, when approximately three quarters of the annual precipitation fall during the summer months from June to September [START_REF] Litt | Glacier ablation and temperature indexed melt models in the nepalese himalaya[END_REF][START_REF] Perry | Precipitation characteristics and moisture source regions on Mt. Everest in the Khumbu, Nepal[END_REF]. For the year 2019-2020, the Northern Bay of Bengal is the most important source of moisture during the monsoon (62 %), more than from the Indus River Delta and Arabian Sea (21 %) and Indo-Gangetic Plain (17%) [START_REF] Perry | Precipitation characteristics and moisture source regions on Mt. Everest in the Khumbu, Nepal[END_REF]. From October to May, the westerlies on the continent are predominant for precipitation [START_REF] Perry | Precipitation characteristics and moisture source regions on Mt. Everest in the Khumbu, Nepal[END_REF]. The monsoon usually stops suddenly at the end of September/beginning of October when the weather, mostly cloudy, calm and warm with frequent precipitation switches within a few days to constantly sunny and dry conditions, progressively colder and windier, during the post monsoon (October-November). Very dry, sunny and windy conditions prevail during the winter (December -February) where there is almost neither any clouds nor precipitation. From March to May, the pre-monsoon is characterized by a progressive increase in convective clouds bringing warm and humid air, with progressively more and more precipitation and a decrease in strong winds [START_REF] Bookhagen | Topography, relief and TRMM-derived rainfall variations along the Himalaya[END_REF][START_REF] Sherpa | Contrasted surface mass balances of debris-free glaciers observed between the southern and the inner parts of the Everest region[END_REF][START_REF] Wagnon | Reanalysing the 2007-19 glaciological massbalance series of mera glacier, Nepal, central Himalaya, using geodetic mass balance[END_REF].

2-3 Processes controlling the glacier mass balance

In this Indian monsoon-dominated region, over debris-free glaciers, ablation is dominated by melt in the lower part of the glacier below the equilibrium line altitude (ELA) above which sublimation plays a substantial role during all seasons, except the monsoon (e.g. [START_REF] Wagnon | Seasonal and annual mass balances of Mera and Pokalde glaciers (Nepal Himalaya) since 2007[END_REF][START_REF] Stigter | The Importance of Snow Sublimation on a Himalayan Glacier[END_REF][START_REF] Litt | Glacier ablation and temperature indexed melt models in the nepalese himalaya[END_REF]Fugger et al., 2022). The radiation budget mostly controls the melt [START_REF] Stigter | Energy and mass balance dynamics of the seasonal snowpack at two high-altitude sites in the Himalaya[END_REF]. The incoming shortwave radiation is maximal during the pre-monsoon, before the installation of permanently overcast conditions during the monsoon, responsible for maximal incoming longwave radiation compensating the heat loss via outgoing longwave radiation [START_REF] Bonekamp | Contrasting Meteorological Drivers of the Glacier Mass Balance Between the Karakoram and Central Himalaya[END_REF][START_REF] Matthews | Going to Extremes: Installing theWorld's HighestWeather Stations on Mount Everest[END_REF]. Overall, the net all-wave radiation is high during the pre-monsoon and monsoon and is controlled by the albedo and by the seasonality of cloudiness (the percentage of cloud in a given atmospheric layer) [START_REF] Bonekamp | Contrasting Meteorological Drivers of the Glacier Mass Balance Between the Karakoram and Central Himalaya[END_REF][START_REF] Stigter | Energy and mass balance dynamics of the seasonal snowpack at two high-altitude sites in the Himalaya[END_REF]. Snow with its impact on albedo and air temperature and its control on the rain-snow limit, are therefore very important meteorological variables influencing the glacier mass balance in the ablation area. Wind speed, which governs sublimation, is more important in the accumulation area. Most of the ablation still happens during the monsoon due to stronger melt than during the pre-monsoon season, when refreezing compensates the melt until the cold content of the glacier surface layers originating from the winter has been evacuated (e.g., Fugger et al., 2022).

3-Methods

3-1 Field sampling

The field campaigns took place in 2014, 2016 and 2018. Moraines were sampled and mapped in the field using a Garmin GPS survey instrument (precision 10 m) (Table 1). Using a hammer and a chisel, 51 granitic boulders (> 60 cm in height) were sampled from the crest of the 14 selected moraines and on roches moutonnées (bedrock) close to the front of the glaciers. A Suunto Compass Clinometer PM-5 was used to measure topographic shielding for each sample in the field. Sample elevation was extracted from the handheld Garmin GPS instrument. All the boulders were photographed and their ground-to-sample height was measured. We have considered well-preserved moraines (> 1 m high and > 20 m long), and selected boulders and locations on the moraine with no evidence of disturbance caused by the action of other processes (river, rock fall or debris flow; [START_REF] Jomelli | Comparing characteristics of rockfall talus and snow avalanche landforms in an alpine environment using a new methodological approach[END_REF].

In order to document the deglaciation chronology of Mera (ME) and South Khare (SK) glaciers, we first collected samples on roches moutonnées close to their current front position (Figs 2345). Other samples were collected downslope on lateral moraines at approximately 1-2 km from the current front.

3-2 Sample preparation and AMS measurement

All samples were processed at CEREGE (Aix-en-Provence, France). Samples were crushed and sieved to collect the 250-1000 µm fraction. Quartz was first concentrated by magnetic separation and then isolated by successive leaching in a ⅔ H2SiF6/ 1/3 HCl mixture. The obtained quartz fraction was leached at least 3 times in a 10% HF -10% HNO3 solution, in order to remove any remaining feldspars and to clean the grains from atmospheric 10 Be. Purified quartz was completely dissolved in concentrated HF (48%) after addition of an in-house 9 Be carrier solution (3025 ± 9 ppm; Merchel et al., 2008). Beryllium was extracted by successive alkaline precipitations of Be (OH)2 alternated with separation on anion and cation columns. Samples were then oxidized at 700°C for 1 hour and the final BeO mixed with Nb powder and loaded into copper cathodes. AMS measurements of the 10 Be/ 9 Be ratios were conducted at the French national AMS facility ASTER [START_REF] Arnold | The French accelerator mass spectrometry facility ASTER: improved performance and developments Nuclear Instrumentation Methods in Physics Research Section B: Beam Interactions with Materials and Atoms[END_REF]. Samples were calibrated against the inhouse standard STD-11 ( 10 Be/ 9 Be = 1.191 ± 0.013 *10 -11 ; [START_REF] Braucher | Preparation of Aster in-house 10be/9be standard solutions[END_REF] and a 10 Be halflife of 1.387 ± 0.0012 * 10 6 years [START_REF] Chmeleff | Determination of the 10Be half-life by multicollector ICP-MS and liquid scintillation counting[END_REF][START_REF] Korschinek | A new value for the half-life of 10 Be by Heavy-Ion Elastic Recoil Detection and liquid scintillation counting[END_REF]. Analytical uncertainties combine ASTER counting statistics, standard uncertainty, external uncertainty (0.5%; [START_REF] Arnold | The French accelerator mass spectrometry facility ASTER: improved performance and developments Nuclear Instrumentation Methods in Physics Research Section B: Beam Interactions with Materials and Atoms[END_REF] and machine blank correction. 10 Be concentration in the sample was calculated from the corresponding 10 Be/ 9 Be ratio and was corrected from associated chemical blanks (Table 1). Scaling to the sample locations was made according to the recent, physicallybased, LSD model [START_REF] Lifton | Scaling in situ cosmogenic nuclide production rates using analytical approximations to atmospheric cosmic-ray fluxes[END_REF] which performs similarly to older empirical models presented in [START_REF] Borchers | Geological calibration of spallation production rates in the CRONUS-Earth project[END_REF]. Chosen parameters include the ERA 40 atmospheric reanalysis [START_REF] Uppala | The ERA-40 re-analysis[END_REF] and the VDM 2016 geomagnetic database. We used the SLHL global production rate of 4.08 ± 0.23 atoms g -1 yr -1 as no regional production rate is available so far.

Exposure ages were computed with the online CREp program [START_REF] Martin | The CREp program and the ICE-D production rate calibration database: a fully parameterizable and updated online tool to compute cosmic-ray exposure ages[END_REF] http://crep.crpg.cnrs-nancy.fr) and are presented in Table 1. In order to make our results easily comparable to previous studies carried out in other regions of the HMA, we present in Table 2 the exposure ages computed using the time-dependent "Lm" scaling model [START_REF] Lal | Cosmic ray labeling of erosion surfaces: in situ nuclide production rates and erosion models[END_REF][START_REF] Stone | Air pressure and cosmogenic isotope production[END_REF] and recalibrated previous data when comparisons were made with previous studies. 10 Be CRE ages are reported with 1σ "external" uncertainties, which include measurement, production rate and scaling uncertainties (e.g., [START_REF] Balco | A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10Be and 26Al measurements[END_REF], in the main text and on the maps (Figs. 2,4) for better comparison with other proxies. Both internal and external uncertainties are reported in Table 1. For comparison we have recalculated all moraine ages using the different tests such as Chevenet's and Peirce's referenced in [START_REF] Batbaatar | Asynchronous glaciations in arid continental climate[END_REF] that are summarized in the table S1. However, for a given moraine, its assigned age corresponds to the weighted mean of the sample ages that successfully passed a Chi² test (calculated with the internal uncertainties) used to identify outliers [START_REF] Ward | Procedures for comparing and combining radiocarbon age determinations: a critique[END_REF]. Chi² test was preferred because it is the most conservative test compared to the others (see Table 1). Based on field observations, we also considered the stratigraphic relationships to identify outliers.

3-3 Paleoclimate data from the two transient model simulations

To explore links between glacier behavior and climate forcings we have used output from two available transient global climate model (GCM) simulations from LOVECLIM (Kobashi et al., 2016) and CCSM4 [START_REF] Liu | Transient simulation of deglacial Climate Evolution with a new mechanism for Bolling-Allerod warming[END_REF]. The so-called TraCE simulation from CCSM4 includes variations in insolation, greenhouse gas concentrations, ice sheet extension and altitude from the ICE5-G reconstruction and also a crude estimate of associated freshwater release in the North Atlantic. LOVECLIM uses similar forcings, but ICE-6G for the ice sheet, and does not account for freshwater release in the North Atlantic. This second transient simulation also includes an estimate of volcanic forcing, which is not included in the TraCE simulation.

As the Atlantic Meridional Overturning Circulation (AMOC) evolution is poorly represented in the two models, we corrected the transient simulations using the [START_REF] Ayache | Multi-centennial variability of the AMOC over the Holocene: A new reconstruction based on multiple proxy-derived SST records[END_REF] reconstruction of the AMOC.

The AMOC reconstruction for [START_REF] Ayache | Multi-centennial variability of the AMOC over the Holocene: A new reconstruction based on multiple proxy-derived SST records[END_REF] was then calibrated using Northern Hemisphere reconstruction of [START_REF] Kaufman | A global database of Holocene paleo temperature records[END_REF] and transient model simulations as well as hosing simulations from [START_REF] Swingedouw | Decadal fingerprints of freshwater discharge around Greenland in a multi-model ensemble[END_REF], in order to quantify the climatic impacts of AMOC variations. From there, the variables simulated in the transient run have been corrected using a semi-empirical model described in detail in [START_REF] Jomelli | AMOC control on millennial-scale glacier changes during the Holocene[END_REF].

For comparative purposes we also included in our analysis Mid-Holocene to present-day evolution of the ISM from other transient global simulations described in detail in Cretat et al., (2020). Here we benefited from sensitivity analyses conducted by the authors who investigated the effects of horizontal resolution, land hydrology, vegetation and runoff on changes in the ISM, with a focus on its seasonality at the monthly time scale and on extremes at the seasonal time scale. They also analyzed changes in the interannual-to-decadal variability of the ISM from two model versions of the IPSL Earth System model.

3-4 Comparison with other moraine chronologies from the HMA

To get a better understanding of glacier fluctuations in the HMA we explored other CRE moraine records from the worldwide Ice-D database (http://alpine.ice-d.org/). Here we have applied four criteria to be consistent with the analysis. We did not consider moraines dated with only one CRE age or unpublished data. We focused on glacial valleys without undated moraine that may have been formed during the timeframe investigated in this study. As an example, a study that would provide an age for a Lateglacial moraine without dating other moraines located upstream would not be conclusive for our purpose and thus would not be considered in the regional comparison. In addition, a moraine dated with ages that show a large dispersion is rejected for the same reason as possibly reflecting inheritance or post depositional changes. Following the same strategy, we did not consider glacial valleys with moraines dated with uncertainties that do not make possible to attribute the moraine formation to a specific period (Lateglacial or Early, Mid or Late Holocene).

4-Results

We present 51 10 Be CRE ages in this contribution, 25 10 Be CRE ages from Mera glacier and 26 10 Be CRE ages from South Khare glacier.

4-1 Mera glacier chronology

The chronology of the Mera glacier is based on samples from roche moutonnées located just a few meters below the current front position of the glacier and from six moraines (Figs. 2,3).

Four samples (D1, D2, D3, D4) were collected on roche moutonnées that were covered by ice until recent years. Apparent CRE ages of these samples range between 0.05 ± 0.03 ka to 0.25 ± 0.09 ka. As these surfaces were covered by ice and therefore protected from cosmic rays, until recent years (this is evidenced in recent aerial photos and successive field trips during which we witnessed the frontal retreat of the glacier) their apparent ages should be null. The apparent young ages for D1 and D4 are probably due to inheritance due to insufficient resetting by ice abrasion since a potential previous exposure or pre-exposure under reduced ice thickness during the last centuries. Five samples (B1, B5, B7, 17nep11 and 17nep12) were taken on this youngest and highest ridge ME-M1. 10 Be CRE ages are between 0.06 ± 0.02 ka (B7) and 0.43 ± 0.08 ka (B1). Among these samples, B7 was considered as an outlier based on Chi 2 test and was excluded from the mean age calculation of the moraine that yields a 10 Be CRE age of 0.30 ± 0.03 ka (n=4). Three samples were collected on the Mera right lateral moraine ME-M2. Here we selected the largest boulders and tried to avoid boulders that could have fallen from the moraine ME-M1. These samples B2 (0.75 ± 0.13 ka), B3 (0.90 ± 0.25 ka), and B4 (0.72 ± 0.13 ka) yield a weighted mean 10 Be CRE age of 0.8 ± 0.1 ka (n=3). On the third nested right lateral moraine ME-M3, the two samples 17nep10 (1.31 ± 0.25 ka) and 17nep13 (1.71 ± 0.3 ka) yield a mean 10 Be CRE age of 1.5 ± 0.2 ka. The right lateral moraine ME-M4 was dated at 11.0 ± 0.3 ka from the weighted mean of six samples (I1, I2, I3, J1, J2, J3) that showed rather consistent 10 Be CRE ages ranging between 10.7 ± 0.6 ka and 11.6 ± 1 ka. ME-M5 was dated to 2.3 ± 0.2 ka based on three samples (17nep15, 17nep16 and 17nep17). Finally, the older short ridge left lateral moraine ME-M6 was tentatively dated to 11.3 ± 0.6 ka based on the only sample 17nep20, because 17nep19 age (0.30 ± 0.05 ka) is considered as an outlier based on stratigraphy.

4-2 South Khare glacier chronology

Samples were collected on eight moraine ridges and recently ice-free roches moutonnées close to the front of the glacier (Figs. 4,5). The four samples (C1, C2, C3 and C4) on these roches moutonnées show apparent ages from 0.12 ± 0.04 ka (C3) to 0.4 ± 0.07 ka (C1). Here again these roches moutonnées were covered by ice until recently suggesting some inheritance related to a complex history of the glacier over at least the Holocene or a reduced thickness of ice over the last centuries.

On the crest of moraine SK-M1 of South Khare glacier, 3 samples 17nep5, 17nep7, and 17nep8 were collected on the uppermost part of the ridge. 17nep7 dated to 0.87 ± 0.08 ka is considered as an outlier based on the Chi 2 test and therefore rejected from the mean age calculation of SK-M1, that gives a weighted mean 10 Be CRE age of 0.36 ± 0.03 ka. At a distance of about 15 m horizontally from the right lateral moraine on the upper ablation area, another lateral ridge is visible towards the north west of the main valley. This right lateral moraine SK-M2 was dated to 2.9 ± 0.3 ka based on two samples (17nep6 and 17nep9). North of this landform, two nested crests forming the large vegetated moraine located on the right side of the glacier were investigated. On the youngest one three large blocks (1-2 m in length) (17nep2, 17nep4 and Mera15) were selected near the upper ablation zone, where the slope of the longitudinal profile of the moraine is gentle. These samples yield a mean weighted 10 Be CRE age of 9.7 ± 0.4 ka. At the base and on the proximal slope of SK-M3 two samples H1 (10.49 ± 0.58 ka) and H2 (11.25 ± 0.75 ka) were collected on a remnant ridge of about 100 m long and about 3m in relief. These small blocks (70 -90 cm long) yield a mean 10 Be CRE age of 10.8 ± 0.5 ka (SK-M4; n= 2).

Assuming this ridge to be connected to the SK-M3 and considering H2 as an outlier (based on the Chi 2 test), the moraine ensemble SK-M3 and SK-M4 would give a mean 10 Be CRE age of 9.9 ± 0.2 ka n=4). Finally, three samples were collected on large boulders (> 1m) on the remnant ridge SK-M5 (14nep1, 17nep1 and 17nep3) which show a weighted mean 10 Be CRE age of 13.6 ± 0.5 ka (n=3).

On the left side of the glacier two samples were collected on the uppermost ridge SK-M6 (Mera7 and Mera9) that yield a mean 10 Be CRE age of 0.24 ± 0.04 ka (n=2), which is a bit younger than the youngest crest SK-M1 on the right size of the glacier. On the distal slope of the other lateral and discontinuous lateral crest SK-M7 dated to 2.8 ± 0.6 ka (n=2) may be synchronous with SK-M2 on the right side of the glacier (dated to 2.9 ± 0.3 ka). This oldest lateral SK-M8 moraine on the left side of the glacier was dated to 12.8 ± 0.5 ka based on the mean of 3 samples (17nep22, G1 and G2).

5-Discussion

5-1 Evolution of Mera and South Khare glaciers since the Lateglacial

The 51 10 Be CRE ages from samples collected on moraines and roches moutonnées from Mera and South Khare glaciers in the upper Dudh Koshi basin help addressing questions about the Lateglacial and Holocene glacier fluctuations and their related climate conditions in the central Himalaya. The oldest documented moraines are dated to ~13 ka (the two oldest moraines at the base of South Khare glacier SK-M5 13.6 ± 0.5 ka; SK-M8 12.8 ± 0.5 ka). SK-M5 may have been formed during the Bølling/Allerød (B/A) (between about 14.7 and 12.9 ka) while SK-M8 may have been formed earlier or during the Younger Dryas (YD) (defined between 12.87 ± 0.3 and 11.61 ± 0.4 ka; [START_REF] Cheng | Timing and structure of the Younger Dryas event and its underlying climate dynamics[END_REF] on South Khare glacier. Field observations suggest that the glacier experienced an almost comparable extent in the Early Holocene (11.6 -8 ka) as revealed by the existence of the nested moraine SK-M3 (9.7 ± 0.4 ka) on the right side of the glacier against SK-M5 and thus was deposited by a glacier of comparable extent. Since no moraine was dated from the end of the Lateglacial so far on Mera glacier, we speculate that the Early Holocene glacier advance experienced by the glacier is also its oldest moraine record over the investigated period. Thus, the behavior of the two glaciers at the end of the Latelacial remains elusive and more data are needed to better constrain their evolution during the transition between the warm B/A and the YD.

During the Holocene we evidenced that the largest glacial extent occurred in the Early Holocene at ~ 9-11 ka on the two glaciers (ME-M4 dated to 11.0 ± 0.3 ka and SK-M3 dated to 9.7 ± 0.4 ka). In addition, we did not find any other moraine formed during the Mid-Holocene (~8-4 ka) suggesting the extent of the two glaciers was smaller during this period than during the Late Holocene (~3-0 ka). Moreover, apparent ages of roche moutonnées recently uncovered by ice showed some inheritance or potential accumulation through the ice that may corroborate this hypothesis. During the Mid-Holocene the front of the glacier would have been located at a higher altitude than the current altitude favouring 10 Be accumulation. During the Late Holocene larger glacier advances than during the Mid-Holocene would have eroded earlier pre-exposed bedrock surfaces but not totally removed 10 Be accumulated during these previous Mid-Holocene periods free of ice. One may also hypothesize that 10 Be might have accumulated when ice thickness was reduced to a few meters due to energetic muon particles [START_REF] Dunai | Cosmogenic Nuclodes. Principles, concepts and application in the Earth surface Sciences[END_REF]. In such a hypothesis the retreat of the glacier during the Mid-Holocene or periods during the Late Holocene could have just been slightly smaller than the latest Late Holocene glacier advances which caused larger extents evidenced by the dated moraines reported in this study. From about 3 ka, several advances were recorded with at least two advances larger than those occurring during the Little Ice Age on both glacier's margin [START_REF] Rowan | The 'Little Ice Age' in the Himalaya: a review of glacier advance driven by Northern Hemisphere temperature change[END_REF][START_REF] Solomina | Glacier fluctuations in the last 2000 years[END_REF].

The oldest one may have occurred at about 2.5 ka as suggested by ME-M5 dated to 2.3 ± 0.2 ka on Mera glacier's site, SK-M2 dated to 2.9 ± 0.3 ka and SK-M7 dated to 2.8 ± 0.6 ka on South Khare glacier's site. Another glacier advance period may have occurred at the base of Mera glacier at about 1.5 ka as reported by ME-M3 dated to 1.5 ± 0.2 ka. Finally several advances occurred during the last millennium with ME-M2 dated to 0.8 ± 0.1 ka, and others occurring during the Little Ice Age with SK-M1 dated to 0.36 ± 0.03 ka, ME-M1 dated to 0.3 ± 0.03 ka, and SK-M6 Dated to 0.24 ± 0.04 ka.

Interestingly, the moraines of the investigated glaciers did not vary synchronously at a centennial time scale but revealed a common pattern of change at a millennium time scale. We suspect that these differences in the chronology at a multidecadal time scale may be due to the morphology and topography of the glacier, moraine preservation and uncertainties associated with the CRE ages [START_REF] Gibbons | Probability of moraine survival in a succession of glacial advances[END_REF][START_REF] Brun | Heterogeneous influence of glacier morphology on the mass balance variability in high mountain Asia[END_REF]. In addition, on the two glaciers, the different extents recorded during the Holocene, in particular during the Late Holocene, have a rather similar amplitude.

5-2 Comparison with other glacier chronologies in Nepal

To make easier comparisons, previous chronologies mentioned below were recalculated following the same cosmogenic approach as the one presented in the method section of this paper (production rate, scaling scheme, atmospheric pressure etc). The two small debris-free glaciers investigated in this study revealed intriguing evolution compared with the evolution of another debris-free Paldor glacier located in Ganesh region, central Nepal [START_REF] Gayer | Monsoonal forcing of Holocene glacier fluctuations in Ganesh Himal (Central Nepal) constrained by cosmogenic 3 He exposure ages of garnets[END_REF].

We notice that the most significant phase of retreat of the debris-free Paldor glacier in Ganesh region (central Nepal), occurred before the Early Holocene (Gayer et al., 2006) (Figs.1, n°5 on fig. 6). This rapid reduction in ice volume was followed by a relative stability of Paldor glacier extent around 11 ka. A second phase of retreat starting soon after 8.5 ka, with a minor readvance during the recent centuries, is suggesting comparable glacier behavior in Ganesh and Everest regions.

On the other hand, in the Khumbu valley, two moraines formed by the debris-covered glacier flowing from Hillary peak in between the Nuptse and Lhotse summits are dated to the Last Glacial Maximum [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF] defined between 26.5 ka and 19 ka [START_REF] Clark | The last glacial maximum[END_REF] and the Late Holocene respectively. These results show that the maximum Holocene ice extent of this debris-covered glacier occurred during the Late Holocene. This suggests that this glacier would have another pattern of change compared to the debris-covered Khumbu glacier (with a maximum extent dated to ~7 ka) and the two debris-free glaciers investigated in this study.

However, more data are needed before being conclusive as undated moraines located in between the Last Glacial Maximum and the Late Holocene exist (this is the reason why we excluded such a case with undated moraines for our comparison). In addition, the age of two moraines downslope from the debris tongue at the base of the northern face of Ama Dablam (Fig. 1; n°7 on Fig. 6) investigated by [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF] reveals that the glacier formed a massive moraine during the YD-Early Holocene transition (~12-11 ka) and another one during the Late Holocene respectively (n°7 on Fig. 6). Thus, the debris-covered tongue of Ama Dablam glacier would show a potentially similar evolution at a millennial time scale with the two debris-free glaciers investigated in this study if one assumes the massive moraine to be formed during the Early Holocene. Finally, the Chhukung stage moraine (defining Early Holocene advance) at the base of the debris-covered Khumbu glacier (n°9 on Fig. 6; [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF] shows ages from the YD using the same calculation criteria (LSD, VDM, ERA 40 and the same production rate). However, recent investigations on the lateral moraines of Khumbu (n°9 on Fig. 6) suggest a maximum Holocene extent during the Mid-Holocene which is asynchronous with that of the adjacent Lobuche glacier (n°8 on Fig. 6) occurring during the Late Holocene [START_REF] Hornsey | Be-10 dating of ice-marginal moraines in the Khumbu Valley, Nepal, Central Himalaya, reveals the response of monsoon-influenced debris-covered glaciers to Holocene climate change[END_REF]. Thus, despite this remarkable pioneer work conducted about 20 years ago in the Everest region [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF] and more recent investigations, it remains difficult to compare the evolution of debris-free and debris-covered glaciers as more data are needed to be conclusive. Nevertheless, some moraines of the Late Holocene observed on Mera and South Khare glaciers were also documented on the two Khumbu and Lobuche glaciers [START_REF] Hornsey | Be-10 dating of ice-marginal moraines in the Khumbu Valley, Nepal, Central Himalaya, reveals the response of monsoon-influenced debris-covered glaciers to Holocene climate change[END_REF] during the wet season (JJA) from TraCE full forcing experiment [START_REF] Liu | Transient simulation of deglacial Climate Evolution with a new mechanism for Bolling-Allerod warming[END_REF]. c). Monsoon reconstruction from a compilation of stalagmite δ 18 0 from 14 caves (Yu et al., 2022). d).

Monsoon reconstruction from marine core MD 177-176 in the Gulf of Bengal after [START_REF] Marzin | Glacial of the indian monsoon and their relationship with north atlantic climate: New data and fluctuations d modelling experiments[END_REF]. e). Monsoon reconstruction from SO188-342KL in the Gulf of Bengal after [START_REF] Contreras-Rosales | Evolution of the Indian Summer Monsoon and terrestrial vegetation in the Bengal region during the past 18 ka[END_REF]. f). Monsoon reconstruction from sediment core SK 148-55 in Arabian Sea after [START_REF] Thamban | Indian Summer Monsoon Variability during the Holocene as Recorded in Sediments of the Arabian Sea: Timing and Implications[END_REF]. g). Annual precipitation reconstruction from a sediment core in Sikkim region after [START_REF] Ali | High frequency abrupt shifts in the Indian summer monsoon since Younger Dryas in the Himalaya[END_REF]. h). and i). AMOC reconstructions from [START_REF] Thornalley | Long-term variations in Iceland-Scotland overflow strength during the Holocene Geoscientific Instrumentation Methods and Data Systems long-term variations in Iceland-Scotland overflow strength during the Holocene[END_REF] and [START_REF] Ayache | Multi-centennial variability of the AMOC over the Holocene: A new reconstruction based on multiple proxy-derived SST records[END_REF]. j). 10 Be CRE age probability density distributions with their analytical uncertainties during the Holocene from mountain glaciers of the investigated Mera (purple shaded curves) and South Khare glaciers. Grey vertical rectangles show major chronozones.

5-3 Comparison with other glacier chronologies in the HMA

Comparing Mera and South Khare glacier chronologies with other glaciers in the HMA does not provide robust conclusions [START_REF] Finkel | Beryllium-10 dating of Mount Everest moraines indicates a strong monsoonal influence and glacial synchroneity throughout the Himalaya[END_REF][START_REF] Owen | Quaternary glaciation of Mount Everest[END_REF]Saha et al., 2019) . Indeed, glacial chronologies covering the investigated Lateglacial-Holocene period without undated moraine that may have been formed during this time remain very rare (n<10).

Moreover, these rare cases do not exhibit a clear regional pattern whatever the type of glaciers considered (debris-free or debris-covered) with a specific glacier evolution in dominated westerlies or monsoonal regions. For instance, in Garhwal Himalaya, glaciers from Bhagirathi valley (n°2 Fig. 6, [START_REF] Barnard | Style and timing of glacial and paraglacial sedimentation in a monsoon-influenced high Himalayan environment, the upper Bhagirathi Valley[END_REF] may have an in-phase evolution (maximum extent in the Early Holocene) with debris-free glaciers reported in this study. However, glaciers in Kashmir, or in Langtang Himal [START_REF] Barnard | Landscape response to deglaciation in a high relief, monsoon-influenced alpine environment, Langtang Himal, Nepal[END_REF][START_REF] Murari | Timing and climatic drivers for glaciation across monsoon-influenced regions of the Himalayan-Tibetan orogen[END_REF]Saha et al., 2018) (Figs.1,6), would have another long-term Holocene evolution with a maximum extent asynchronous with that reported in this study. As mentioned earlier, current observations suggest a strong mass balance variability driven by the respective influence of precipitation and temperature on glacier mass balance [START_REF] Sakaï | Contrasting glacier responses to recent climate change in high mountain Asia[END_REF][START_REF] Brun | A spatially resolved estimate of high mountain Asia glacier mass balances from 2000 to 2016[END_REF]. Here we hypothesize that the strong heterogeneity of glacier fluctuations in the past reflects the current heterogeneity of glacier sensitivity to climate conditions as well as the spatial variability of climate change at the scale of the Asian continent. 5-4 An unresolved relationship between climate forcings and the observed trend of the two debris-free glaciers during the Holocene Multi centennial glacier fluctuations documented by 10 Be moraine records reported in this study revealed a possible large glacial extent during the Bølling/Allerød (at the base of SK glacier) and two other large glacier advances, recorded on both glaciers, during the Early and Late Holocene, respectively. The formation of SK-M5 during the B/A corresponds to an enhanced period of precipitations reported by TraCE model and a recent compilation of stalagmite δ 18 O data from a total of 14 caves (Yu et al., 2022) spanning 0°-35°N, 70°-95°E, that establish a continuous integrated ISM proxy record. However on both records this does not correspond to the maximum of precipitation that occurred during the Early Holocene instead (Fig. 7).

Consequently, it remains unclear why SK-M5 is the largest advance while it did not occur during the maximum of precipitation and why we did not observe such an advance on Mera glaciers. Thus we will not further explore possible climate conditions that would explain the formation of this moraine. However, we will discuss the possible climate forcings during the Holocene that may cause the observed large extension during the Early Holocene and the second large advance period occurring in the Late Holocene. For the present-day, the two debris-free glaciers investigated in this study are driven during the wet season on the one hand by precipitation from the Bay of Bengal and on the other hand by temperatures that impact the altitude of the snow/rain phase [START_REF] Litt | Glacier ablation and temperature indexed melt models in the nepalese himalaya[END_REF][START_REF] Wagnon | Reanalysing the 2007-19 glaciological massbalance series of mera glacier, Nepal, central Himalaya, using geodetic mass balance[END_REF]. We will thus focus successively on these two potential driving factors throughout the Holocene using climate model results and proxy data.

It is generally considered that a strong ISM occurs during summer insolation maxima and causes enhanced precipitation to the Himalaya (Rachid et al., 2011). By contrast, reduced precipitation related to a weak monsoon occurs during cold periods and weak summer insolation. In a simple way these two large advance periods during the Early and Late Holocene should be explained by an increase in precipitation compared with that during the Mid-Holocene during which these two glaciers are supposed to have retreated.

The evolution of precipitation simulated by the full forcing experiments revealed after a maximum in the Early Holocene a progressive decrease throughout the rest of the Holocene (Fig. 7b). None of TraCE and LOVECLIM model simulations show precipitation variations that perfectly fit with the two glacier moraine records investigated in this study. While a maximum ISM in the Early Holocene is concomitant with our moraine records, the simulated evolution of climate conditions during the Mid and Late Holocene cannot simply explain glacier evolution. Causes for this discrepancy can be multiple and might notably be linked to biases in the models. Indeed, from a multi-configuration ensemble of up-to-date fully coupled global transient simulations covering the last 6 ka, Cretat et al (2020) evidenced several modeldependent issues. Tests conducted over the 20 th century show that the vegetation-climate interactions modulate the timing of the monsoon onset in the model which can be shifted by more than two months compared to observations. The location of the 2 mm/day isohyet from April to September shows remarkable differences between observations and their simulations. Moreover, differences between observations and their simulations in the variability of monsoon rainfall can exceed 30%. Over the last 6 ka, the drying trend from the Mid-Holocene including extremes, and a contraction of the rainy season evidenced in the IPSL model is larger in simulations with higher horizontal resolution of the atmosphere and improved land surface hydrology. In addition, Mid to Late Holocene changes in JJAS rainfall mean-state documented in the IPSL model show an extraordinary spatial variability with regions where the drying trend is more significant than others. As precipitation is a key driver of the investigated glaciers, we explore the reconstructions of paleo-precipitation of the ISM to see if they agree with climate model simulations and can explain the glacier evolution. Unfortunately, terrestrial and marine paleo-precipitation records are rare and do not perfectly correlate with model results and moraine records (Rachid et al., 2011). For instance, MD77-176 core located in the Bay of Bengal (14°310 N, 93°080 E, Fig. 1) [START_REF] Marzin | Glacial of the indian monsoon and their relationship with north atlantic climate: New data and fluctuations d modelling experiments[END_REF] indicates enhanced monsoon rains over the Indian subcontinent between 11-8 ka followed by a progressive decrease during the rest of the Holocene (Fig. 7d), in contrast with the general pattern from TraCE simulation (Fig. 7b). A similar pattern of change is shown from a sediment core SO188-342KL (19°.97 N, 90°.03 E;[START_REF] Contreras-Rosales | Evolution of the Indian Summer Monsoon and terrestrial vegetation in the Bengal region during the past 18 ka[END_REF] in the Bay of Bengal.

Hypothesizing that the current major source of precipitation falling on Mera glacier from the Bay of Bengal may not be stationary through time [START_REF] Ali | High frequency abrupt shifts in the Indian summer monsoon since Younger Dryas in the Himalaya[END_REF], we thus explored other oceanic and terrestrial monsoon precipitation reconstructions from the Arabian Sea and India respectively (Grupta et al., 2005;[START_REF] Ali | Holocene Indian Summer Monsoon variability from the core monsoon zone of India, a pollen-based review[END_REF][START_REF] Banerji | Holocene climate variability and indian summer monsoon: an overview[END_REF]. However again, none of their results agree with our paleo moraine records (see for instance the sediment core SK148-55 from Thamban et al., 2007 Fig. 7f). Thus, the maximum ISM in the Early Holocene documented both in the composite record of stalagmite δ 18 O data from a total of 14 caves (Fig. 7c) (Yu et al., 2022) and in the models agrees with the maximum glacier extent, while there would be a divergence for the rest of the Holocene. In addition, a remarkable exception comes from reconstructed precipitation from a glacial outwash sedimentary profile in Chopta valley, Sikkim Himalaya (located east of the Khumbu region). Here [START_REF] Ali | High frequency abrupt shifts in the Indian summer monsoon since Younger Dryas in the Himalaya[END_REF] showed humid periods that roughly match with our moraine records (Fig. 7g). Indeed, the authors identified two periods of unstable climate conditions with higher precipitation during the Early (between 10.6-8 ka) and Late Holocene (between 2.9-0.3 ka) respectively, interrupted by dry conditions during the Mid-Holocene. Furthermore, the authors inferred, in agreement with some recent studies (e.g. [START_REF] Sun | Infuence of Atlantic meridional overturning circulation on the East Asian winter monsoon[END_REF][START_REF] Marzin | Glacial of the indian monsoon and their relationship with north atlantic climate: New data and fluctuations d modelling experiments[END_REF], that the ISM precipitation reconstruction was linked to variations in the North Atlantic climate on multi centennial to millennial time scales, with the AMOC being a potential driver of these changes in North Atlantic climate and its teleconnection with the ISM (e.g. [START_REF] Marzin | Glacial of the indian monsoon and their relationship with north atlantic climate: New data and fluctuations d modelling experiments[END_REF].

As AMOC forcing is known to interact with monsoon (IPCC SROCC 2019), we thus investigated the possible influence of this AMOC forcing on our moraine records. Interestingly, recent reconstruction of AMOC changes revealed two weak AMOC periods during the Early and Late Holocene respectively (Thormaley et al., 2013;[START_REF] Ayache | Multi-centennial variability of the AMOC over the Holocene: A new reconstruction based on multiple proxy-derived SST records[END_REF]. To further explore this possible influence of AMOC on the ISM and more generally, on the two debrisfree glaciers' evolution during the investigated period, we first extracted regional precipitation series at an annual time scale and during the wet season (JJA) from the two full forcing transitory simulations that cover the Holocene simulated by CCSM4 TraCE and LOVECLIM climate models. As AMOC is poorly represented in these two models during the Holocene [START_REF] Jomelli | AMOC control on millennial-scale glacier changes during the Holocene[END_REF], we used AMOC reconstructions to compute corrected GCMs precipitation changes over the Holocene according to the method described in Jomelli et al. (2022) (Fig. 8). AMOC corrected monsoon precipitation changes show strong variations with a progressive increase in precipitation culminating during the Mid-Holocene followed by a progressive decrease. Interestingly, these AMOC-corrected precipitation changes (Fig. 8 blue curves) show a good correspondence with precipitation from proxy records in the Bay of Bengal (Fig. 7d,e). This precipitation pattern of the AMOC-corrected precipitation is in line with various modeling results, showing a stronger ISM when the AMOC and the North Atlantic is warmer (e.g. [START_REF] Jackson | Global and European climate impacts of a slowdown of the AMOC in a high resolution GCM[END_REF]. However, this pattern does not show a good correspondence with our moraine records since it is going in the opposite directions to what might be required to explain precipitation-driven increase in the glaciers, (i.e. an increase in precipitation).

Temperature variations would be superimposed on the precipitation evolution. We may also consider that the impact of the two variables (precipitation and temperature) on the glacier would be opposite stronger monsoon (positive influence on mass balance) associated with warmer temperature (negative influence on mass balance) while the influence of temperature could be compensated by an increase in precipitation. However, here again none of TraCE and LOVECLIM model simulations show temperature variations that perfectly fit with the two glacier moraine records investigated in this study (Figs 7,8).

All in all, the relationships between climate conditions and long-term glacier changes in the investigated sector of Nepal remain puzzling, and reveal potential issues in the modeling of climate conditions, which could be related to the regional climatic responses in this mountainous region. For instance, due to extreme topography, a large spatial variability of precipitation is currently observed in the central Himalaya, leading to very different precipitation trends over short distances; this is still not captured by models. Moreover, the mass balance sensitivity is highly dependent on the total annual precipitation and therefore may have changed over the Holocene. Finally, the major source of precipitation falling on Mera glacier has potentially changed over time, making it impossible to find a clear relationship between regional climate change and local glacier evolution. To overcome such issues, teleconnections between AMOC and climate conditions in Mera region need further investigations as dynamic downscaling in particular, but this remains beyond the scope of this study. In addition, new moraines chronologies from debris-free glaciers across the mountain range may help to identify a regional trend confirming or negating the evolution of Mera and South Kare glaciers.

6-Conclusion

We documented the evolution of the two debris-free Mera and South Khare glaciers during the end of the Lateglacial and the Holocene in the central Himalaya from 51 cosmogenic 10 Be surface exposure ages of samples collected from boulders on moraines and from roche moutonnées. Mera glacier is of particular interest because an energy and mass balance monitoring has been conducted since 2007 enabling a deeper understanding of glacier climate relationships over the recent period. These instrumental measurements indicate that the glaciers' mass balances are driven during the wet summer season (i) by precipitation from the Bay of Bengal and (ii) by temperatures that impact the altitude of the snow/rain phase.

Our paleo glaciological investigations revealed that South Khare glacier evidenced a large glacier extent during the Bølling/Allerød. But glacier advance during this period remains puzzling and new records are needed to better understand climate drivers. On both glaciers 10 Be CRE ages also revealed the existence of Early Holocene moraines and attest of their maximum size at that time over the last 11.6 ka. Moreover, the lack of moraine dated to the Mid-Holocene and 10 Be inheritance for some roche moutonnées at the base of the current ice front positions suggest the two glaciers were smaller during the Mid-Holocene than during the Late Holocene.

From about 3 ka several advances were recorded, which were larger than the LIA.

Current observations suggest both glaciers were driven on the one hand by ISM precipitation, and on the other hand by temperature. During the Holocene, these two variables might have been sensitive to summer insolation changes and AMOC changes among other factors. Strong spatial precipitation variability is also evidenced suggesting possible differences between moraine records and proxies or regional precipitation changes estimated from climate models. However, we did not see any ISM precipitation and temperature changes documented from most proxy records and climate model simulations that would perfectly explain the reconstructed glacier changes presented in this study. If the maximum glacial extent is concomitant with enhanced ISM documented from models and some proxy records, the Late Holocene glacier advance remains puzzling. We, therefore, claim that new glaciogeomorphological data combined with improved GCMs output are necessary to get a better understanding of glacier climate relationships in the past in this region.

  e. Mera ME-M1 to M6 (Figs. 2, 3). A large terminal moraine over 200 m wide formed by the nested crests (ME-M1, -M2, -M3), is cut by a small river from the glacier with a frontal ramp 50 m in relief (Figs 2, 3). On the right side of the Mera glacier about 400 m downstream from the front and about 200 m below in altitude, two large lateral moraines are visible. The inner one is composed of the three nested ridges (ME-M1, -M2, -M3). The ME-M1 lateral moraine is continuous over about 1.5 km from the terminus and is actually superimposed on two other lateral moraines ME-M2 and ME-M3 and certainly explains its large relief. The ME-M1 moraine has unvegetated or poorly vegetated surfaces and unstable proximal slopes with crest heights up to 80 m above the proglacial sandur formed by the river flowing from the front of the glacier. The Mera ME-M2 and ME-M3 right lateral moraines are fragmented and only identifiable from 200 m and 500 m respectively up-glacier from the terminus. Both crests are separated by 10-50 m horizontally from the Mera moraine ME-M1 and at a lower altitude (30 and 50 m respectively). The distal slope of these two moraines is steep but stable and vegetated. The relief of these two ridges is not well developed and generally does not exceed 3 m in height. Distant to about 200 m from the rampart towards the north, separated by approximately 50 m horizontally from ME-M3, an older lateral moraine (ME-M4) is totally covered by vegetation.

Figure 1 .

 1 Figure 1. Location map of Mera and South Khare glaciers (blue frames) and other relevant

  Down-valley of the right side of the glacier, two distinct crests (SK-M3, -M5) were investigated. The youngest crest forms a continuous right lateral moraine SK-M3 of about 300 m in length. The relief of this lateral moraine exceeds 50 m. The steep proximal slope is dissected by relict and active gullies while the distal slope shows only very few boulders with a sub horizontal surface. A remnant ridge SK-M5 of 2 m in relief on the right side of the glacier is distant to about 10 m from SK-M3 horizontally. This discontinuous ridge is only visible near the ablation zone while downslope the crest connected to the lateral moraine SK-M4 disappears. On the left side of the glacier two crests are also visible on the upper part of the ablation area close to each other horizontally The uppermost ridge SK-M6 forms a continuous crest of about 50 m in relief with a symmetric cross profile. The altitude of this ridge between 5000 and 5150 is the same as SK-M1 on the right side of the glacier. This moraine is up to 80 m in relief dominating the village of Khare. On the distal slope of the ridge another discontinuous crest SK-M7 is visible at about 5040 m a.s.l. over a distance of 50 m. Approximately 150 m horizontally towards the south from SK-M7 an old vegetated and rounded moraine SK-M8 of about 30-50 m in relief is visible.

Figure 2 .

 2 Figure 2. Chronology of the Mera glacier's evolution. Geomorphological features are mapped

Figure 3 .

 3 Figure 3. Pictures from Mera catchment and moraines. a). Terminal Late Holocene moraine;

Figure 4 .

 4 Figure 4. Chronology of the South Khare glacier's evolution. Geomorphological features are

Figure 5 .

 5 Figure 5. Pictures from South Khare catchment and moraines. a). Roche moutonnée at the base

Figure 6 .

 6 Figure 6. Some glacial chronologies in Himalaya and summer precipitation. a). The

Figure 7 .

 7 Figure 7. Nepalese glacier chronologies compared to climate proxies. a). Summer insolation

Figure 8 .

 8 Figure 8. Estimates of temperature and precipitation in Mera region from the two GCMs TraCE

Acknowledgments

The ASTER AMS national facility (CEREGE, Aix-en-Provence) is supported by the INSU/CNRS, the ANR through the "Projets thématiques d'excellence" program for the "Equipements d'excellence" ASTER-CEREGE action and IRD. Field work, and mass and energy balance results have been supported by the French Service d'Observation GLACIOCLIM, part of IR OZCAR. This work would not have been possible without the support of the IRD International Joint Lab WATER-HIMAL (PIs: D. Shrestha, and P. Wagnon)

and all the efforts from people in the field: porters, students, helpers, colleagues who are greatly acknowledged here. T. de Garidel and F. Bassino are also warmly acknowledged for their help with ISM data reconstructions.

Yu, Z., Colin, C., Wilson, D. J., Bayon, G., Song, Z., Sepulcre, S., Dapoigny, A., Li, Y., Wan, S., 2022. Millennial variability [START_REF] Lifton | Scaling in situ cosmogenic nuclide production rates using analytical approximations to atmospheric cosmic-ray fluxes[END_REF] which performs similarly to older empirical models [START_REF] Borchers | Geological calibration of spallation production rates in the CRONUS-Earth project[END_REF]. Chosen parameters include the ERA 40 atmospheric reanalysis [START_REF] Uppala | The ERA-40 re-analysis[END_REF] and the VDM 2016 geomagnetic database.

Competing interests

The authors declare no competing interests Declaration of Competing Interest (.docx) Author contributions V.J. designed the paper. V.J., PW., F.B., and D.S, realized the field Sampling R.B., J.C., A.H., S.B and V.J. processed the 10Be sample preparation and analysis. V.J., R.B., J.C., A.H., S.B and ASTER TEAM computed the cosmogenic nuclides ages. D.S. ran the AMOC simulations. V.J., D.S., and P.W. analyzed forcings. V.J. wrote the first draft of the paper and all authors contributed to the discussion and final version of the manuscript.


Senior Member, IEEE Emmanuel Boutillon

Senior Member, IEEE Chris Winstead

Senior Member, IEEE Fakhreddine Ghaffari

The Syndrome Bit Flipping Algorithm for LDPC Codes

Keywords:

Performance of LDPC decoders at high SNR is dominated by trapping sets that induce an error floor in the performance curve. We propose a new algorithm that resolves trapping sets and lowers the error floor. The new algorithm, called Syndrome Bit Flipping (SBF), computes the sum of adjacent parity violations at each symbol node. Bits are flipped by comparing the syndrome sum against a time-varying threshold called the decoding key. SBF is compared to other bit-flipping decoders on the Binary Symmetric Channel (BSC), and is demonstrated as a post-processing step for a Noisy Gradient Descent Bit-Flipping (NGDBF) hardware decoder. We demonstrate the post-processing method for an LDPC code defined in the 802.3an standard, and find that the frame error rate is improved by at least two orders of magnitude, even as the required iterations are reduced by 33%.

I. INTRODUCTION

During the past two decades, Low Density Parity Check (LDPC) codes have emerged as a crucial component of many communication standards, and are increasingly important in memory technologies. One of the main challenges for LDPC decoders is to reduce or eliminate the error floor. To achieve this, many methods of post-processing have been developed, which can reduce the floor by up to a few orders of magnitude. In this paper, we consider a method of post-processing that is especially suited to bit-flipping algorithms. The new algorithm, called Syndrome Bit Flipping (SBF), ignores channel information and considers only the parity-check syndrome states for the final iterations of bit flipping. Algorithm diversity is obtained by varying a bit-flipping threshold in each postprocessing iteration.

In order to motivate the syndrome-only strategy, we first consider an analogy to compressed sensing. We then show that a threshold sequence, which call the decoding key, can be chosen to resolve dominant trapping set errors that are commonly associated with LDPC error floors. This method of post-processing lowers the error floor in bit-flipping decoders, and costs minimal hardware overhead since it only requires masking part of the existing operations in bit-flipping decoders. There is some cost of extra iterations needed for postprocessing, however our simulations show that post-processing E. Boutillon is with Lab-STICC, UMR 6285 CNRS, Université Bretagne Sud, 56100 Lorient, France. Email: emmanuel.boutillon@univ-ubs.fr C. Winstead is with the Department of Electrical and Computer Engineering, Utah State University, Logan, UT, 84322 USA. Email: chris.winstead@usu.edu F. Ghaffari is with the ETIS Lab, ENSEA, CY Cergy Paris University, France. Email: fakhreddine.ghaffari@cyu.fr Manuscript received February 2023 allows using fewer iterations of the primary decoder, so the total iterations are actually decreased while still obtaining better performance. The remainder of the paper is organized as follows: Sec. II-B describes the motivation, notation, and steps of the proposed syndrome-only decoding algorithm. Sec. III presents a detailed analysis of the algorithm for common trapping sets that are known to be dominant contributors to error floors in standard LDPC codes. In Sec. IV we present simulation results of the algorithm as a post-processing solution for Noisy Gradient Descent Bit Flipping (NGDBF) [START_REF] Sundararajan | Noisy gradient descent bit-flip decoding for LDPC codes[END_REF], Probabilistic Gradient Descent Bit Flipping (PGDBF) [START_REF] Rasheed | Fault-tolerant probabilistic gradient-descent bit flipping decoder[END_REF], [START_REF] Le | A probabilistic parallel bit-flipping decoder for lowdensity parity-check codes[END_REF] and their variant algorithms [START_REF] Savin | Gradient descent bit-flipping decoding with momentum[END_REF] simulated on additive white Gaussian noise (AWGN) and Binary Symmetric Channel (BSC), respectively. Finally, Sec. V offers discussion and conclusions.

Notation: In the sequel, binary variables and loop indexes are represented by lowercase letters. Integer variables are represented by capital letters, integer algorithm parameters by Greek letters, and matrices by bold-face capital letters.

II. SYNDROME BASED DECODING ALGORITHM

A. Compressed Sensing Analogy

In the classical theory of block codes, it is well known that when a single error is present in a received data frame, the location of the error corresponds to a unique syndrome pattern that can be obtained via a simple binary calculation. For some codes, this concept can be extended to correct multiple errors. Such a concept is not readily available for modern LDPC codes, where there can be a larger number of initial errors in the received frame. We can, however, make a weaker argument based on compressed sensing: when the number of errors is small relative to the code's frame length, then the syndrome pattern most likely corresponds to a unique error pattern. In compressed sensing, a vector v of dimension n with only a few non zero components, say t with t n, can be compressed into a vector w of size m = t(1 + ), with 1, by multiplying v with a random matrix R of size

(m, n): w = R v.
Compressed sensing is generally performed with real or complex numbers. Nevertheless, the theory is applicable to coding theory where operations are done over a Galois Field. Let us consider the IEEE 802.3an LDPC code defined for the 10 Gbit/s Ethernet Standard [START_REF]IEEE Standard for Ethernet[END_REF]. This LDPC code is defined by a parity-check matrix H with size 2048 × 384. It is a regular code with check degree d c = 32 and variable degree d v = 6. By definition, any codeword c verifies H c = 0. Now let us assume that the codeword is received through a BSC. The received symbols are d = c ⊕ e where e is a binary vector of length n, with t non null components (i.e. t errors of transmission). Thus, the syndrome s = H d mod 2 is equal to s = H( c ⊕ e) = H e and depends only on the error pattern e.

To give a more exact example of this analogy, suppose a given received codeword of the (2048, 1664) 802.3 LDPC code with t = 20 errors and an associated syndrome with 60 nonnull components. The number N e of possible error patterns with t = 20 is

N e = 20 2048 = 6.31 × 10 47 , (1)
while the number of syndromes N s with 60 non-null values among m = 384 is equal to

N s = 60 384 = 1.01 × 10 71 . (2)
Since N s N e , the set of error patterns is quite sparse relative to the set of syndrome patterns. Then one can consider, as in the case of compressed sensing, that the knowledge of s is enough to characterize entirely the error pattern e when the number of errors t is small enough (say lower than 20 for example).

This observation motivates us to search for new types of decoding algorithms based only on the syndrome, ignoring the received samples y in the decoding process, except for the initial computation of the syndrome s. In the case of post-processing, we suppose that a soft-information decoder, operating in the error-floor region, is applied to the received samples y, and produces a vector d of binary decisions. In the error floor region, the decoder's output should have at most a relatively small number of errors corresponding to some absorbing set. We propose that these residual errors should be correctable using a syndrome-only calculation.

B. The SBF Algorithm

The Syndrome Based Flipping algorithm can be defined as a variant of the Gradient Descent Bit Flipping (GDBF) algorithm, with two fundamental differences. First, the channel information is no longer taken into consideration. Second, the flipping decision is based on a predefined threshold θ( ) that depends on the iteration number only. In principle, SBF can be used for post-processing with any LDPC decoding algorithm; in this paper we consider SBF in relation to GDBF due to its close similarity and straightforward hardware adaptation.

To describe the algorithm's steps, we first briefly restate the main points of the NGDBF algorithm [START_REF] Sundararajan | Noisy gradient descent bit-flip decoding for LDPC codes[END_REF]. Given received sample vector y, a symbol-wise energy function is computed. The NGDBF energy function can be expressed as

E (NGDBF) k = -αx k y k + j∈M k s j + q k , (3)
where xk ∈ {+1, -1} is the bipolar decision at iteration , M k is the set of parity-checks adjacent to xk , s j ∈ {0, 1} are the corresponding binary parity syndrome values, q k is an artificial noise perturbation, and α is a scale factor. The decision xk is flipped if E k exceeds a flipping threshold θ.

In simplest terms, the SBF algorithm modifies the energy function by considering only the adjacent parity information:

E (SBF) k = j∈M k s j . (4)
E (SBF)
k is equal to the number of non-satisfied checks adjacent to variable x k . The corresponding bit is flipped if E k > θ( ), where 0 ≤ θ( ) ≤ max (d v ) (similar to the original parallel M-GDBF algorithm [START_REF] Wadayama | Gradient descent bit flipping algorithms for decoding LDPC codes[END_REF]). The precise steps of the SBF algorithm are detailed in Algorithm 1. When SBF is used for postprocessing in bit-flipping algorithms, the hardware cost is quite low. This is demonstrated in Fig. 1 for the case of NGDBF, where a global mode-select signal is routed to mask the channel sample and noise by forcing q k := 0 and y k := 0.

III. TRAPPING SET ANALYSIS

It is now well known that error floors of LDPC codes are affected by cycle subgraphs known as absorbing sets or trapping sets, and for BP-based decoders there are many approaches for mitigating them, e.g. [START_REF] Han | Low-floor decoders for LDPC codes[END_REF]. It was previously shown that decoder diversity -dynamic algorithm rules that change from one iteration to the next -can be exploited to resolve trapping set errors and improve performance of low-complexity decoding algorithms, e.g. in the framework of Finite Alphabet Iterative Decoders (FAID) [START_REF] Planjery | Finite alphabet iterative decoders part I: Decoding beyond belief propagation on the binary symmetric channel[END_REF], [START_REF] Declercq | Finite alphabet iterative decoders part II: Towards guaranteed error correction of LDPC codes via iterative decoder diversity[END_REF]. In this

s j θ - q k × y k × × sgn D
x k mode Fig. 1. Modified symbol node with dual-function capability for both NGDBF and SBF. The added "mode" input is a binary signal that selects NGDBF when mode = 1, and SBF when mode = 0.

section, we examine decoder diversity in SBF by way of the dynamic flipping threshold. We show that a sequence of threshold values can be devised to incrementally resolve known dominant error patterns due to trapping sets.

A. Isolated trapping sets

We first consider trapping sets in isolation, ignoring their neighborhoods within a larger code. For a each particular trapping set, we obtain a specific threshold sequence that corrects all symbol nodes within the set, independent of the initial condition. We refer to such a sequence as a decoding key for the trapping set. We do not offer a proof that such sequences exist for all trapping sets. Instead they are discovered through exhaustive search, and sequences are reported in supplemental data for 29 different trapping set topologies [START_REF]The Syndrome Bit Flipping Algorithm for LDPC Codes[END_REF].

As a first example, a (3,3) trapping set is shown in Fig. 2 for three different error configurations. If all bits are initially in error (Fig. 2.a), then the local parity sum will be E = 1 at each of the symbol nodes in the set. If the first flipping threshold is set to θ(1) = 0, then all erroneous bits are flipped.

If only two symbols are in error in the (3,3) trapping set (Fig. 2.b), then all symbols will have a parity sum E = 2. If the second threshold is θ(2) = 1, then all bits are flipped, resulting in a single-error condition where the correct symbols have E = 1 and the erroneous symbol has E = 3. To correct this final case (Fig. 2.c), the third threshold is adjusted to θ(3) = 2, so that the erroneous bit is corrected.

A more generalized analysis is obtained by analyzing the error state transition diagram for the trapping set. One example is shown in Fig. 3, where the states represent error patterns within the trapping set subgraph, with 000 being the errorfree state. The edges in Fig. 3 are labeled with the value of θ( ) that will induce a transition from one error pattern to another. From the state transition diagram, we deduce a set of Θ sequences guaranteed to reach state 000 regardless of the initial state. In this example, there are six sequences of length three which guarantee correction regardless of the initial state:

Θ 1 = {0, 1, 1}, Θ 2 = {0, 1, 2}, Θ 3 = {1, 1, 0}, Θ 3 = {1, 2, 0}, Θ 5 = {2, 0, 1} and Θ 6 = {2, 0, 2}.
Multiple trapping sets can be resolved if the dominant trapping set graphs are known. For each known trapping set, the transition analysis is performed to obtain decoding keys that resolve each trapping set type. The keys are concatenated to produce a complete key for the code. Since trapping sets are small graphs, it is straightforward to enumerate all paths, increasing the path length until the shortest decoding keys are discovered. This method was used to find solutions for 29 trapping sets selected from a public dataset [START_REF]Trapping set ontology[END_REF], and the resulting decoding keys are provided as supplemental data.

B. Heuristics for non-isolated trapping sets

We say that a trapping set is non-isolated if the SBF algorithm causes error propagation in the set's neighborhood. Errors propagate outside the trapping set neighborhood whenever θ ( ) = 0. The problem is illustrated in Fig. 4, which shows that the neighbors of an erroneous symbol node all have E = 1, so they will be flipped to erroneous states whenever θ = 0. If a θ sequence contains repeated zeros, then errors tend to propagate to a large neighborhood.

Heuristic 1: Whenever θ ( ) = 0, θ ( + 1) > 0 and θ ( + 2) > 0. Rationale: Simulations indicate that in most instances, propagated errors are corrected if the zero threshold is followed by one or more non-zero threshold values in the decoding key.

Heuristic 2: Error propagation is suppressed by inserting θ = d v -1 and θ = d v -2 into the decoding key. Rationale: A solitary error (i.e. an erroneous symbol with no adjacent errors) may be produced as a consequence of error propagation, and is corrected when θ = d v -1. Similarly, an adjacent pair of erroneous symbols is corrected when θ = d v -2.

Heuristic 3: SBF is only applied for bits with "unreliable" channel samples, |y| ≤ γ for some threshold γ. The choice of γ is determined empirically by simulation. Rationale: This heuristic bounds the neighborhood in which errors may propagate, thereby improving the probability that all errors can be resolved.

IV. PERFORMANCE EVALUATION

A. SBF Performance on BSC

We first evaluate the SBF algorithm on the BSC as a standalone decoding algorithm, and evaluate its performance in comparison to other bit-flipping algorithms for the IEEE 802.3an LDPC code. The error floor for this code is dominated by an [START_REF] Planjery | Finite alphabet iterative decoders part I: Decoding beyond belief propagation on the binary symmetric channel[END_REF][START_REF] Planjery | Finite alphabet iterative decoders part I: Decoding beyond belief propagation on the binary symmetric channel[END_REF] trapping set pattern. The decoding key for this simulation was prepared using Heuristics 1 and 2, and is provided online as a supplemental file. The simulation results are shown in Fig. 5. At frame error rates above 10 -6 , standalone SBF performance is somewhat worse than PGDBF or BP. The improvement of SBF is seen in the error floor region, where the frame error rate approaches 10 -8 . In this performance domain, SBF performs better than either PGDBF or BP, likely because of its resistance to dominant trapping set errors that affect the other algorithms.

B. SBF as Post-Processing on AWGN

SBF was implemented as a post-processing option for the NGDBF algorithm, and tested on two codes. Decoding keys were generated applying Heuristics 1 and 2 as described in Section III. The first code is the IEEE 802.3an LDPC code, simulated using FPGA emulation of a 5-bit quantized AWGN channel. The results are shown in Fig. 6. The figure includes quantized BP (5-bit) and unquantized BP, using data from Zhang et al. [START_REF] Zhang | Design of LDPC decoders for improved low error rate performance: quantization and algorithm choices[END_REF]. NGDBF was found to reach an error floor similar to quantized at an FER of 10 -6 . With SBF postprocessing, the error floor is lowered to about 10 -8 , similar to unquantized BP.

× = 1 × E = 1 E = 1 (a) × E = 2 × E = 2 E = 2 (b) × E = 3 E = 1 E = 1 (c)
The results also show that the total iterations can be significantly reduced when using SBF, requiring a maximum of four hundred iterations for NGDBF+SBF, whereas NGDBF in isolation must allow at least 600 iterations to achieve adequate performance on this code, up to 1000 iterations for best performance [START_REF] Sundararajan | ASIC design of a noisy gradient descent bit flip decoder for 10GBASE-T ethernet standard[END_REF]. By lowering the maximum iterations from 600 to 400, the decoder's worst-case throughput is improved by 33% in addition to the improved error floor.

The final result in Fig. 6 shows results for NGDBF with SBF post-processing where Heuristic 3 is applied. These results were obtained by capturing error frames from the hardware emulation. Those frames were then decoded again in software with NGDBF+SBF applying all three heuristics.

The second code simulated was a larger code with d v = 5 and N = 10240, with results shown in Fig. 7. This code reaches a very high error floor with NGDBF, but the floor is suppressed below an FER of 10 -6 when using NGDBF+SBF.

V. CONCLUSION

We presented a new low-complexity LDPC decoding algorithm suitable for post-processing to lower the error floor. The method has a very simple implementation because it mainly requires disabling a portion of the symbol node calculations. Some overhead is also needed to distribute a global integer threshold value that changes in each iteration. A procedure was given for optimizing the threshold sequence in order to resolve errors on known dominant trapping sets in the code. APPENDIX The set of 17 decoding key sequences S 17 = {Θ k } k=1,2,...,17 used for the results in Fig. 5 are given below. They are sorted in descending order, for clarity. In a real application, the order of execution of each decoding key should be modified to perform first the decoding key sequences that correct the highest number of remaining errors. This smart ordered reduced the average iteration before finding a codeword.

Algorithm 1

 1 Syndrome Based algorithm Input data: Noisy binary codeword d and parity check matrix H. List of S n sequences of thresholds, {Θ k } k=1,...Sn with Θ k = {θ k ( )} =1...N k where N k is the length of the k th sequence. Initialization: Compute s 0 = H d mod 2 ; Set k = 0; Algorithm: while k < S n and s = 0 do = 0 ; k = k + 1 ; s = s 0 ; e = 0; while < N k and s = 0 do = + 1; E = H T s; for i = 1 to N do if E(i) > θ k ( ) then e(i) = 1e(i); end if end for s = H e mod 2; end while end while if s = 0 then e = 0 end if return z = d + e

Fig. 2 .Fig. 3 . 1 E = 1 E = 1 Fig. 4 .

 231114 Fig. 2.Example of trapping set correction under dynamic threshold flipping. Erroneous symbol nodes are indicated by ⊗; degree-two check nodes appear as solid boxes, and degree-one check nodes as unfilled boxes. At each symbol node, the total local parity sum E is indicated. Three states are shown: (a) the initial state with three errors, (b) two errors, and (c) one error.

Fig. 5 .

 5 Fig. 5. Test results obtained from simulation of the rate 0.8413 802.3an code on the BSC, compared to the recent PGDBF algorithm and the standard belief propagation (BP) algorithm.

Fig. 6 .

 6 Fig.6. Test results obtained from an FPGA implementation of NGDBF with post-processing for the 802.3an code. The NGDBF+SBF results use 300 iterations of NGDBF followed by 100 iterations of SBF. The NGDBF+SBF+h3 result indicates that Heuristic 3 was applied.

Fig. 7 .

 7 Fig. 7. Simulation results of NGDBF with post-processing for a regular (5, 20) LDPC code of length N = 10240. NGDBF was used for up to 1000 iterations, followed by 11 post-processing iterations with SBF.

Θ 1 =

 1 {5534324}; Θ 2 = {5454543332}; Θ 3 = {544442543333}; Θ 4 = {544333233323323323323323323143323323}; Θ 5 = {544233242332}; Θ 6 = {5433343233333}; Θ 7 = {5435432433433333323}; Θ 8 = {534434333333}; Θ 9 = {53343333}; Θ 10 = {45433333233}; Θ 11 = {443433322332333}; Θ 12 = {44233}; Θ 13 = {433334233}; Θ 14 = {354333333}; Θ 15 = {34433324332}; Θ 16 = {333332433}; Θ 17 = {323244423}.