Carsten Rohr

Paul Abbott

James P Connolly

Keith W J Barnham
email: k.barnham@imperial.ac.uk

Massimo Mazzer

Lucia Nasi

Chris Button

John S Roberts

Graham Clarke

Ravin Ginige

K Ravin Ginige

InP-based lattice-matched InGaAsP and strain-compensated InGaAs∕ InGaAs quantum well cells for thermophotovoltaic applications

I. INTRODUCTION

In a Quantum Well Cell (QWC) nanometre thick layers of low bandgap material are introduced into the intrinsic region of a p-i-n diode in order to increase the absorption of a photovoltaic (PV) cell [1]. The absorption threshold or effective bandgap can be tuned by varying the material composition of these quantum wells (QWs) and, because of the quantum confinement, by changing their thickness. For cells based on the InP substrate, a wide range of lattice-matched materials is available, In 1-x Ga x As y P 1-y with x ≈ 0.47y, where the bandgap can be varied between 0.9 and 1.7 µm, with In 0.53 Ga 0.47 As having the lowest possible bandgap of this lattice-matched material system.

The absorption can be extended if the QWs are strained to the substrate. But only a few QWs can be grown before dislocations form and the device quality deteriorates due to non-radiative recombination [2]. This restriction can be overcome by strain compensating QWs and barriers. In principle, an unlimited number of periods can be grown without strain relaxation if the stress is minimised by appropriate strain-compensation [3,4].

In thermophotovoltaics (TPV) [5,[START_REF]Special issue on thermophotovoltaics[END_REF], heat radiation is converted into electricity, as opposed to sunlight in a conventional PV cell. The source is at a lower temperature for TPV applications, typically around 1500-2000 K, as opposed to the 5800 K of the sun. The peak of a lower temperature black-body spectrum is at a longer wavelength, hence lower band-gap semiconductors offer a better spectral match. It is also possible to shape the illuminating spectrum, for example by introducing selective emitters between the source and the devices to obtain a narrow band spectrum [7], which increases the conversion efficiency when matched to the cell. Such selective emitters are often based on rare-earth elements, such as Erbium, Thulium or Holmium with peak emission of about 1.5, 1.8 and 2.1 µm, respectively [7].

In this paper we study and analyse lattice-matched InGaAsP QWCs suitable for an Erbium-based emitter (Section III A), and strain-compensated InGaAs/InGaAs QWCs for a Thulium-based emitter (Section III B). A more general review of QWCs for TPV applications is given in Ref. [START_REF] Connolly | Special issue on thermophotovoltaics[END_REF]. Test devices were processed into 1 mm diameter mesas with a contact ring leaving an optical window of 600 µm diameter.

A Keithley 238 Source-Measurement Unit was used for current-voltage (I-V) measurements in the dark and under illumination. The temperature of the sample, which was kept constant at 25 o C using a Peltier plate, was monitored with a Keithley 142 Digital Multi-Meter and a platinum resistance thermometer. An Oriel quartz halogen tungsten lamp was used as light source with a stabilised DC power supply and a fibre lead, approximating a 3000 K black-body spectrum. The power density was set to a constant level using a calibration cell.

The quantum efficiency (QE) was measured using a Bentham M300 0. A. Lattice-matched InGaAsP QWCs

Several lattice-matched InGaAsP QWCs were grown similar to MR1178 (see Table I) and a nominally identical sample MR1180 which had an additional etch-stop layer between the base and the substrate for front contacting. For comparison, control samples without QWs but otherwise the same as in Table I (i.e. with the same total i-region width) were grown: a barrier control MR1088, where the whole i-region consists of the same barrier material Q1.1;

and a well control MR1091 where the i-region consists of a quaternary material composition so that the effective absorption edge is the same as in the QWC (i.e. material with a slightly wider bandgap than Q1.6 to compensate for the confinement in the QWs). Hence samples MR1088 and MR1091 are double hetero-structures. As one can see in Fig. 1, samples MR1091 and MR1180 have exactly the same absorption edge, while MR1178 extends somewhat further. These samples are very well suited for TPV applications with Erbium-based selective emitters which have a peak at about 1500 nm (also indicated in Fig. 1).

One would expect the voltage performance to be determined by the bandgap of the QW material. The the most challenging situation is to compare samples with the same effective threshold, as in this set of samples. Although the absorption threshold is identical for MR1091 and MR1180 (see Fig. 1), the dark current density of the QWC is significantly lower than that of the well control as shown in Fig. 2. The QWC MR1178 shows even better material quality having a similarly low dark current as QWC MR1180 while the absorption is extended to slightly longer wavelengths.

Barnham et al. have demonstrated a voltage enhancement of QWCs compared to their bulk control cells in the lattice-matched material systems AlGaAs/GaAs, GaInP/GaAs, InP/InGaAs [START_REF] Barnham | [END_REF]. However, in those cases the control cells were made of the same material as the QWs, which means that the bulk control cells have slightly lower effective bandgaps due to the confinement in the QWs. Here we demonstrate that a QWC with the same effective bandgap as a bulk control (Fig. 1) shows a lower dark current (Fig. 2) and a voltage enhancement (Fig. 3). This is further evidence that QWCs have a better dark current than expected from their absorption threshold [START_REF] Barnham | [END_REF]. In Fig. 4, this is shown to be true also for the material system In 1-x Ga x As y P 1-y on InP. One might expect the dark current (for example at a typical operating point such as a bias of 0.4 V) of the QWCs to lie on a line connecting the two homogenous bulk controls InP and InGaAs (Fig. 4), including an InGaAs monolithic interconnected module (MIM) [START_REF] Fatemi | Proc. 26th IEEE PV Specialists Conference[END_REF]. However, all the QWCs have a much lower dark current than that by about two orders of magnitude. The dark current depends to some degree on the number of QWs, particularly for a small number of QWs, but here we consider large numbers of QWs as all the lattice-matched QWCs have 60 QWs. Also included in Fig. 4 are double hetero-structures such as MR1091 and MR1088 which are described above. Although, double hetero-structures have a much lower dark current than the homogenous controls, their dark current is not as low as most QWCs. This shows that InGaAsP QWCs have an improved voltage performance and hence a clear advantage over homogenous bulk control cells and even over double hetero-structures.

The I-V characteristics under illumination from the 3000 K black body source are shown for some of the cells in Fig. 3. Although the absorption thresholds of MR1091 and MR1180 are identical, the well control MR1091 has a slightly higher short-circuit current I SC than the QWC MR1180 because it has a slightly higher QE in the range 1100-1450 nm (see Fig. 1) which gives rise to a considerable contribution under the broad illuminating spec- trum. However, this effect should be much less under a narrow band spectrum, such as an Erbia selective emitter, where the QWC MR1180 is expected to outperform the well control MR1091. Even though I sc is smaller for MR1180 than for MR1091, the V oc is significantly higher for MR1180. It shows that the reduced dark current leads to a higher V oc .

The QWC MR1178 shows an improvement in performance over both the barrier control (MR1088) and the well control (MR1091) even under this broad illuminating spectrum (Table III). This is mainly due to an enhancement in V oc compared to the well control, and a much better I sc compared to the barrier control. However, the fill factor of MR1178 is slightly inferior compared to both controls. It should be noted that these numbers are for comparison relative to each other.

The absolute performance can be much improved by employing an appropriate grid design and optimising the front layers (e.g. etching off the cap layer in the optical window) and applying AR coats. In a TPV system, the illuminating spectrum and the spectral response of the cells should be matched to increase the efficiency. The system efficiency can be further improved for example by employing a mirror at the back of the device to reflect photons which have not been absorbed such as low-energy photons back to the source. Such a mirror is particularly advantageous for QWCs because the QE of the QWs is increased as photons have the chance of a second pass through the QWs.

B. Strain-balanced InGaAs/InGaAs QWCs

For TPV applications, there is increasing interest in longer wavelengths selective emitters, for example based on Thulium and Holmium, in order to match lower temperature blackbody spectra and hence achieving higher system efficiencies compared to Erbia and Ytterbia [7,[START_REF] Mazzer | Proc. Society of Automotive Engineers 2000 World Congress[END_REF]. Burning fossil fuels at lower temperature at around 1500 K reduces NO x emissions.

For such applications low band-gap material is required to match the long wavelength selective emitter. However, appropriate and inexpensive substrates of the required lattice constant and band-gap are not readily available, so the lower band-gap material is often strained. If the material exceeds a critical thickness dislocations are introduced which increase non-radiative recombination. In multi quantum well (MQW) systems one can avoid this strain-relaxation by strain-compensating the layers; alternating wells and barriers have bigger and smaller lattice-constants, but each period is lattice-matched to the substrate.

Differences in elastic constants may need to be taken into account to minimise the stress [3,4]. Using this method, it is possible to obtain smaller effective band-gaps in the quantum wells than the lattice-matched material, and hence extend the absorption towards longer wavelengths without introducing excess dislocations. As long as each individual layer remains below the critical thickness, in principle an unlimited number of periods can be grown without introducing any dislocations.

However, this technique is limited by the formation of strain-balance defects [13,14].

They are caused by the onset of undulations (wavy growth) which originate in the tensile barrier layers. The maximum number of defect free periods in an InGaAs/InGaAs straincompensated MQW is proportional to the inverse of the elastic energy density per unit area stored in each period [13,15]. Several strain-compensated samples have been grown successfully, and one of these sample (E1055) is described in Table II. This sample has a high indium content of about 74% in the QWs, in order to extend the absorption to about 1.95 µm [START_REF] Abbott | The Fifth Conference on Thermophotovoltaic Generation of Electricity[END_REF], but the high In fraction also results in a high misfit of about 1.4 %. The barriers, by contrast, were designed to be much wider to reduce the misfit to about -0.47 %. This way undulations and strain-balance defects could be avoid as is evident from the transmission electron micrograph in Fig. 5.

Several strain-compensated QWCs are also included in Fig. 4 showing that even in the case of strain-compensation the dark current density is lower than expected from the absorption threshold. The line of expected performance as indicated in Fig. 4 is extrapolated from the results of lattice-matched bulk devices, as for these absorption thresholds no latticematched bulk material is available. This shows that with strain-compensation one can extent the absorption and at the same time it gives excellent results for strain-compensated QWCs, having similar or better voltage performance than expected based on the trends in the bulk material in Fig. 4.

IV. CONCLUSIONS

We have shown that lattice-matched InGaAsP QWCs are very well suited for TPV applications such as with Erbia selective emitters. QWCs with the same effective bandgap, i.e. lower bulk bandgap, than a bulk control cell show a better voltage performance. Hence a QWC can lower the dark current, enhance the V oc and hence improve the efficiency compared to a bulk control cell. A major advantage of QWCs is that the bandgap can be engineered by changing the well thickness and varying the composition to fit the illuminating spectrum. This is relatively straight-forward in the lattice-matched InGaAsP system, but it can be extended to longer wavelengths by using strain-compensation techniques, achieving bandgaps that cannot be achieved with lattice-matched bulk material. We have shown that even such strain-compensated QWCs have voltage performances that are at least as good as, if not better than expected. Strain-compensated QWCs are promising candidates for TPV applications, for example with Thulia and/or Holmia selective emitters.

  3 m monochromator with a 100 W tungsten lamp as a light source and a Bentham current stabilised filament lamp power supply 505. The photocurrent is measured with a lock-in technique using a Stanford Research Systems SR510 Lock-in Amplifier with a Bentham 218 variable frequency optical chopper at a frequency of about 187 Hz. Two such set-ups were used, one for the visible range (400-1100 nm) with a calibrated Si detector, and another one for the infrared range (850-2000 nm) with calibrated InGaAs (850-1650 nm) and extended InGaAs (1300-2000 nm) detectors from Hamamatsu.

FIG. 1 :

 1 FIG.1: External Quantum Efficiency of QWCs MR1180, MR1178, compared with well control MR1091 (all non-AR). The position of an Erbia peak is also indicated (arbitrary intensity).

FIG. 2 :

 2 FIG.2: Dark current densities of QWCs MR1180, MR1178, compared with well control MR1091.

FIG. 3 :

 3 FIG. 3: I-V curves of QWCs MR1180, MR1178, well control MR1091 and barrier control MR1088 under 3000 K blackbody illumination.

FIG. 4 :

 4 FIG.4: Dark current density at a fixed bias of 0.4 V vs. absorption threshold.

FIG. 5 :

 5 FIG. 5: Cross-sectional transmission electron micrograph of strain-compensated QWC E1055 (see

TABLE I :

 I Sample description of a typical lattice-matched quaternary QWC, in this case sample MR1178. Q1.1 and Q1.6 represent the quaternary InGaAsP material with a bulk absorption edge of about 1.1 and 1.6 µm respectively.

		Layer	Repeats	Thickness [ Å]	Material	Doping	Conc. [cm -3 ]
		Cap/Contact	1	1000	InGaAs	p	3 × 10 18
		Emitter	1	1500	InP	p	8 × 10 17
		Spacer	1	2500	InP	i
	Lattice-	Barrier	60	70	Q1.1	i
	matched	Well	60	100	Q1.6	i
	MQW	Barrier	1	70	Q1.1	i
		Base/Buffer	1	5000	InP	n	3 × 10 18
	Sheffield, UK. A typical sample with a multi-quantum well (MQW) region is given in Table
	I. The layers were grown on epi-ready (100) InP substrates, n-type with sulphur, and
	usually with a misorientation of 0.35 degrees off to (110). The p-and n-type layers are
	InP, whereas the ∼ 1 µm wide intrinsic region is made of the quaternary Q1.1 (where the Q
	value indicates the bulk absorption edge in microns) with a composition of y ≈ 0.36 resulting
	in a bulk absorption edge of 1.1-1.15 µm. Incorporated into this i-region are 60 QWs of
	100 Å width made of the quaternary Q1.6 with a composition of y ≈ 0.86 equivalent to a
	bulk absorption edge of about 1.6 µm, but due to the confinement the effective absorption
	edge is closer to 1.5 µm. A 0.2 µm wide spacer region is introduced between the i-and
	p-region to reduce Zn dopant diffusion into the QWs [9]. The top InGaAs cap layer serves
	for contacting the device. When an anti-reflection (AR) coating is applied, the cap is etched
	off in the optical window.
	Some strain-compensated samples were also grown in Sheffield (prefix 'MR'), but most
	were grown by IQE Ltd, Cardiff, Wales (prefix 'E'), also by MOVPE, such as E1055 shown
	in Table II.

II. EXPERIMENTAL

The lattice-matched InGaAsP devices were grown on InP substrates by metal-organic vapour-phase epitaxy (MOVPE) at the EPSRC National Centre for III-V Technologies in

TABLE II :

 II Sample description of strain-compensated QWC E1055.

		Layer	Repeats Thickness [ Å]	Material	Doping Conc. [cm -3 ]
		Cap/Contact	1	3000	InGaAs	p	1 × 10 19
		Emitter	1	3000	InP	p	2 × 10 18
		Spacer	1	2000	InP	i
	Strain-	Barrier	40	89	In 0.45 Ga 0.55 As	i
	compensated Well	40	80	In 0.74 Ga 0.26 As	i
	MQW	Barrier	40	89	In 0.45 Ga 0.55 As	i
		Buffer	1	200	InP	i
		Base	1	5000	InP	n	1 × 10 18

TABLE III :

 III Light I-V characteristics under 3000 K black-body illumination. All samples are 1 mm diameter mesas with 600 µm diameter optical window and the cap is left on with no AR coating.

	Sample	V oc [V]	I sc [µA]	Fill Factor	Power Output [µW]
	QWC MR1178	0.477	-109	0.699	36
	Well control MR1091	0.455	-108	0.712	35
	QWC MR1180	0.491	-97.0	0.727	35
	Barrier control MR1088	0.681	-59.3	0.803	32

Table II )

 II .

Acknowledgments

This research benefited from the financial support by the UK Engineering and Physical Science Research Council (grant GR/L50471) and the European Commission (grant ERK6-CT-1999-00019).


Kan-Hua Lee

Keith W J Barnham

Benjamin C Browne

James P Connolly

Jessica G J Adams

Rob Airey

Nicholas J Ekins-Daukes

Markus Fuhrer

Victoria Rees

Mathew Lumb

K. -H Lee

A L Dobbin

M Mazzer

J S Roberts

T N D Tibbits

Multiple quantum well top cells for multijunction concentrator solar cells

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Strain-balanced quantum well middle cells provide many advantages over conventional bulk middle cells, including the ability to extend the absorption edge further towards infrared region and the flexibility to optimize the absorption edge for different spectrum conditions [START_REF] Barnham | Nanotechnology for Photovoltaics[END_REF]. QuantaSol Ltd successfully commercializes triple-junction concentrator cells based on strain-balanced quantum well middle cells. Recently, cross-wafer median efficiencies of QuantaSol's production cells have achieved 40%. Theoretical modeling results also show that triple-junction cells with quantum well middle cells have higher energy yields than bulk 3J cells with similar external quantum efficiencies [START_REF] Lumb | Comparing the Energy Yield of (III-V) Multi-Junction Cells with Different Numbers of Sub-Cells[END_REF].

Top cells with multiple quantum wells can further benefit the efficiency of multiple junction solar cells in a number of ways. They make it possible to independently optimize the absorption edge of both top and middle cell to maximize the efficiency. Furthermore, the radiative dominance of the top cell enhances photon coupling between subcells, which makes the multi-junction cell less sensitive to variations in the incoming spectrum and operating temperature [START_REF] Barnahm | Photonic Coupling in Multi-Junction Quantum Well Concentrator Cells[END_REF].

In this paper, we will first present the characterization results of a novel MQW top cell and in particular discuss its radiative efficiency. We then show the results of simulation results of dual MQW triple-junction solar cell efficiencies by considering photon coupling. Then, we will report experiment results on dual-MQW tandem devices and direct measurements of photon coupling from electroand photo-luminescence.

NOVEL MQW TOP CELL

The MQW top cells in this study are grown at the University of Sheffield and QuantaSol. The bulk regions of the top cell are of a conventional p-on-n GaInP design with a GaInP i-region. The shallow quantum wells are novel, as they consist of quaternary GaInAsP in an essentially lattice matched combination which is immiscible in the bulk. We obtain wells of thickness around 4 nm with high material quality under appropriate growth conditions.

Figure 1 shows the measured and modelled [START_REF] Barnham | Nanotechnology for Photovoltaics[END_REF] (or reference Escorial paper see email) external quantum efficiency (EQE) and Figure 2 is measured and fitted dark-IV of the same device. This device has 22 shallow wells. As shown in Figure 1, the absorption edge the top cell is extended to around 740nm. In Figure 2, different contributions of modelled dark-IV are plotted. The level of the radiative contribution to the dark-current is a prediction based on the fit to the EQE in the quantum well as explained in Refs. [START_REF] Barnham | Nanotechnology for Photovoltaics[END_REF] and [START_REF] Barnahm | Photonic Coupling in Multi-Junction Quantum Well Concentrator Cells[END_REF]. The agreement with the data suggests that the device becomes radiatively dominant in the high bias region.

MODELING PHOTON COUPLING

Photon coupling between subcells in a multi-junction device has been discussed [START_REF] Baur | Effects of optical coupling in III-V multilayer systems[END_REF]. However, a MQW top cell can make photon coupling more significant than in conventional bulk multi-junction devices. In a multi-junction cell with an MQW top cell, radiative recombination may dominate over other contributions to the dark current in the top cell as observed in the last section. The top cell emitter, base and tunnel junctions are transparent to the photons radiated by the top cell quantum wells. For this reason, more photons will be absorbed by the bottom cell. As a result, photon coupling needs to be taken into account when predicting the performance of MQW multijunction solar cells.

The calculation of photon coupling is based on the detailed balance approach proposed by Nelson [START_REF] Nelson | Observation of suppressed radiative recombination in single quantum well p-i -n photodiodes[END_REF]. Figure 3 shows the efficiency contour plot of triple junction GaInP/InGaAs/Ge dual-MQW solar cells. The colored contours assume no photon coupling. To reach the optimum efficiency of bulk triple-junction devices, the band gaps of both the top and middle subcells must be lower than the band gaps in lattice-matched top and middle subcells [START_REF] King | 40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells[END_REF]. One conventional approach to achieving lower band gaps is upright metamorphic growth, which can only be extended further by introducing additional defects and dislocations in the device. However, with MQWs in both top and middle subcells, the absorption edge of these junctions can be independently optimized within the dashed rectangle in Figure 3. The blue lines in Figure 3 are iso-efficiency contours calculated by taking photon coupling into account. It indicates that a radiatively dominated MQW multi-junction solar cell can potentially reach a higher efficiency than bulk metamorphic multijunction devices. The wide area inside the 48% blue contour line also suggests that dual-MQW solar cells are less sensitive to variation of the incoming spectrum, which results in more efficient energy harvesting than bulk multijunction devices.

Figure 3 Calculated efficiency contours of dual-MQW triple-junction solar cells assuming bulk QE of 100% and MQW quantum efficiency of 60% in the top cell and 40% in the middle cell and an AM1.5-AOD spectrum(?).

Color contours are calculated efficiencies without considering photon coupling, while blue lines are efficiencies with the effect of photon coupling considered

PHOTON COUPLING MEASUREMENT

The idea of this photon coupling experiment is producing photoluminescence in the top cell at a given applied bias and subsequently using the bottom cell as a photodetector to measure the photons coupled into it. Three terminal dual-junction devices with additional terminals between top and bottom junction were fabricated allowing subcell to be biased independently. These three-terminal devices are InGaP/GaAs dual junction cells with MQWs in each subcell. The polarity of one junction is inverted and only a highly-doped n-type lateral conduction layer is grown between the top and bottom junction.

In this experiment, we use photoluminescence to generate radiative recombination in the top cell. The bottom cell will then absorb the re-emitted photons from the top cell. The top cell is illuminated by a Spectra Physics Millennia V neodymium vanadate laser capable of producing up to 5.5 W at 532 nm. The intensity of the beam is only 10 -11 of its original intensity when it reached the surface the bottom cell. Under the same laser intensity, we first measure the short-circuit current of top cell ITOP, then we keep top cell open and measure the short circuit current of bottom cell. Hence the dark current component coupled into the bottom cell is equal to the ratio ηCOUP=IBOT/ITOP. This test is repeated with different laser intensity. The radiative efficiency of the top cell can also be obtained from ηCOUP if the factors of photons collected by the bottom cell and bottom cell quantum efficiency are known. As a result, ηCOUP should be proportional to radiative efficiency ideally. This method can also be extended to standard twoterminal dual junction devices. We illuminate the twoterminal devices with the same laser and measure the short-circuit current of the device. In this case, the top cell will be operated at a voltage VTOP which is close to its open-circuit voltage because of strong current mismatch between top and bottom cells. The voltage of the bottom junctions becomes -VTOP. If the bottom cell is not seriously shunted and has reverse breakdown voltage lower than -VTOP, the current at -VTOP will be equal to short-circuit current. This can be checked by varying the bias of the dual-junction device under illumination. If the measured current is not constant when the bias is shifted few hundred millivolts from zero point, then the cell is not suitable for this test. The top cell short-circuit current ITOP under the same illumination can be estimated by two methods. The first one is measuring ITOP with the single junction top cell with same layer structure of the dualjunction top cell. The other one is calculating ITOP by measured top cell QE in the dual-junction cell and known laser intensity. This method will slightly underestimate ηCOUP because the operating voltage of the top cell is lower than its open-circuit voltage.

Figure 4 shows ηCOUP against top cell current density of several different devices. For clarity, the results of twoterminal dual-junction devices are plotted in dashed lines. Some key parameters of these test devices are shown in Table 1. This graph shows that the coupling ratio of device#1, #2 and #3 have very similar trends with the calculated radiative efficiency shown in Figure 2, which is expected. However, the coupling ratio of device#4 has steeper increase in the same range. This may be due to different MQW designs or strong dominance of nonradiative recombination in device#4. This experiment will then be used to calibrate the model described in previous section and optimize multijunction devices by considering photon coupling.

Figure 2

 2 Figure 1 Measured MQW top cell quantum efficiency

Figure 4

 4 Figure 4 Photon coupling ratio against top cell current density. The solid line indicates the result of a three

Table 1 Details of the dual-junction devices used in the photon coupling experiment. The emitter and base in this table are all InGaP.

 1

	device	Termi	MQW well	Emitter	Base
		nals	number	(thickness(nm)/doping density
				(cm	-3 ))
	#1	3	20	100/1E18	570/1E17
	#2	2	40	50/undoped
	#3	2	37	50/undoped
	#4	2	15	100 /5E18	1000/1e17


A Bessière
email: bessiere@chimie-paristech.fr

J P Connolly

K W J Barnham

M F Führer

M Lynch

I M Ballard

M Mazzer

D C Johnson

G Hill

J S Roberts

Observation of reduced radiative recombination in low-well-number Strain-Balanced Quantum Well Solar Cells

Absolute electroluminescence and photoluminescence measurements were carried out on strain-balanced quantum well solar cells Over a range of bias, a reduced radiative recombination in the wells was observed compared to a model assuming a constant quasi-Fermi level separation over the device thickness. This was interpreted as a QFLS suppression in the wells relative to the bulk of 18 and 5 meV respectively for the single and 5 well SB-QWSCs, consistent with previous results on strained Single Quantum Well and Double Quantum Well devices. The photoluminescence spectra at open-circuit voltage under illumination in the well agreed with the electroluminescence spectra in the

light in contrast to some theoretical predictions. Generation of hot carriers in the wells could be the thermodynamically compensating phenomenon for the QFLS reduction.

3 When a p-n junction solar cell is illuminated and/or electrically biased the Fermi level splits into electron and hole quasi Fermi levels over the depletion region of the device. The quasi-Fermi level separation (QFLS) determines the radiative recombination losses of the solar cell. Araujo and Marti [1] used a detailed balance treatment to show that in an ideal cell, the QFLS is equal to the applied bias at all points so that in the radiative limit, no higher efficiency could be achieved by a Quantum Well Solar Cell (QWSC) compared to a conventional cell of optimum band-gap. However studies carried out on strained Single Quantum Well (SQW) [2] and Double Quantum Well (DQW) [3] devices concluded that the QFLS was reduced in the wells compared to the bulk indicating promising efficiency enhancements. Since then the strain-balance concept has been introduced allowing the incorporation of a higher number of wells into the GaAs p-n junction while maintaining the materials crystalline quality [4]. Compressive strain in the low band gap In x Ga 1-x As well material is compensated by tensile strain in the high band gap GaAs 1-y P y barrier material. In this paper we extend the work on QFLS studies from strained well devices to Strain-Balanced Quantum Well Solar Cells (SB-QWSCs).

Three SB-QWSC wafers were grown by MOVPE (Metal Organic Vapour Phase Epitaxy) as p-i-n structures with 1, 5 and 10 quantum wells (QWs) inside the intrinsic region. The emitter and base consisted of C-doped GaAs and Si-doped GaAs respectively. All cells were designed to have the same i-region width (~ 7800 Å). The iregion was grown with two buffer layers of intrinsic GaAs of various thickness either side of 1, 5 or 10 In 0.163 Ga. 0.837 As 95Å-thick well layers. The compressive strain of these wells is compensated by 145 Å-thick GaAs 0.911 P 0.089 barrier layers in tensile strain, separating the wells.

Photodiode structures were processed from the wafers and the spectral response of the three cells have been measured and compared to the in-house simulation model SOL described in Ref. 5. The model solves transport equations for the minority carriers in the charge neutral regions of the cell and calculates the well absorption spectrum from first-principles calculations of the confined states in the well. By adjusting the minority carrier parameters of neutral regions and the QW dimensions in the i-region, one can obtain the best fit of the spectral response and hence set the absorption coefficients of every layer. These absorption coefficients are used by the model to predict the luminescence response of the cells.

The emission of a SB-QWSC is calculated using a detailed balance treatment, explained in detail in Ref. 2 based on the following expression for the emitted light flux

U rad : .dS.dE dΩ ) E s)b(E, , a(E, U 0 S F rad ∫ ∫ ∞ ∆ = θ (1)
where a(E,θ,s) is the probability that a photon of energy E will be emitted or absorbed from the point s on the surface S of the emitting volume at an angle θ and b(E, ∆E F ) is the spectral photon flux density due to spontaneous emission at that point. ∆E F is the local QFLS. The flux density b(E,∆E F ) is defined from the generalized Planck equation for spontaneous emission [6] as:

1 e E c h 2n ) E b(E, )/kT E (E 2 2 3 2 F F - = ∆ ∆ -q (2)
where n is the refractive index of the cell, k is Boltzmann's constant, T the absolute temperature, h Planck's constant and c the speed of light. The spatial geometry of the cell is taken into account in calculating the forward emitted flux [START_REF] Bessière | Proc. 20 th European Photovoltaic Solar Energy Conf[END_REF] which is then compared to experimental electroluminescence spectra.

To be able to compare the absolute intensities of the spectra, the experimental EL was calibrated using the method described in Ref. 8. Low temperature photocurrent and photoluminescence measurements were taken as a function of bias. The reverse bias and the photoluminescence spectra were compared with the forward bias photoluminescence (PL) spectra so that the latter could be expressed as the number of photons emitted per eV and per s.

Experimental calibrated EL spectra were taken at various forward biases for each sample. Examples at +0.87V (a) and +0.92V (b) for the single well, +0.88V for the 5 well (c) and +0.87V for the 10 well (d) SB-QWSCs are shown as circular data points in Fig. 1 where they are compared to the model (broken lines) with a QFLS equal to the applied bias.

For bulk cells with no quantum wells, for example a GaAs p-i-n cell [START_REF] Connolly | Proc. International Conference on Solar Concentrators for the Generation of Electricity or Hydrogen (ICSC-4)[END_REF] the model fits the calibrated EL closely in absolute units assuming the QFLS is equal to the experimentally applied bias throughout the cell.

For quantum well samples however the model assuming constant QFLS throughout, overestimates the well luminescence. This is seen most clearly at the main well excitonic luminescence peak at 1.285 eV for the single and the 5 well samples. The bulk peak is also overestimated due to a contribution from the quantum well, as the GaAsP barrier is higher than bulk GaAs. We find that in order to fit the EL overall, and the main excitonic peak in particular, the model requires a reduced QFLS in the wells of 18, 15 and 5 meV as shown by the full lines in Fig. 1.a, b and c. The single well result is similar to the QFLS reduction of 23 meV observed in a GaAs/InGaAs strained SQW of similar well depth [2] and to the (11-18) meV reduction observed in the deeper well of AlGaAs/GaAs lattice matched DQWs over a similar voltage range [3].

Single well and 5 well SB-QWSCs therefore demonstrate similar results to strained SQW and unstrained DQWs in that radiative losses are less than predicted by assuming a constant quasi-Fermi level equal everywhere to the applied bias which is an assumption made in some models of radiatively dominated QW solar cells [1,[START_REF] Luque | [END_REF]. This QFLS reduction will lead to a significant solar efficiency enhancement if the dominant contribution to dark currents is radiative as has been observed in SB-QWSCs at high concentration [START_REF] Barnham | Proc. 19th European Photovoltaic Solar Energy Conference and Exhibition[END_REF].

Luque and Marti [START_REF] Luque | [END_REF] have used a local thermodynamic argument to suggest that if the QFLS in a QW cell is reduced under dark conditions as observed here, then the QFLS and hence the radiative recombination will be increased under illumination. We have pointed out that the QFLS reduction could be explained by a compensating thermoelectric effect [START_REF] Conibeer | Proc. 20 th European Photovoltaic Solar Energy Conf[END_REF]. An increase of the temperature of the carriers photogenerated in the wells of only a few degrees relative to the crystal temperature (hot carriers) could create a thermal gradient that would balance the gradient of the electrochemical potential [START_REF] Führer | Proc. 3 rd Photovoltaic Science Applications and Technology Conference[END_REF][START_REF] Connolly | Proc. International Conference on Solar Concentrators for the Generation of Electricity or Hydrogen (ICSC-4)[END_REF]. Our observed reduction in QFLS with well number is consistent with this picture [START_REF] Conibeer | Proc. 20 th European Photovoltaic Solar Energy Conf[END_REF]. It has further been suggested that light concentration may amplify the phenomenon by promoting the creation of hot carriers [START_REF] Conibeer | Proc. 20 th European Photovoltaic Solar Energy Conf[END_REF].

For this reason, and to test the Luque-Marti conjecture, we have investigated the behavior of these SB-QWSCs under illumination. The three samples were illuminated with a tunable Titanium:Sapphire laser operating at 1.355 eV, i.e. carriers were excited by illumination in the QW absorption continuum. A holographic notch filter centered on 907 nm was placed in front of the monochromator slit to attenuate the laser line.

Photoluminescence (PL) spectra (full lines) obtained at open circuit voltage (V OC ) are

shown in Fig. 2 together with EL spectra (scattered points) obtained in the dark at the same bias. The EL and PL spectra are identical for all three samples. Hence the radiative recombination spectra and integrated radiative-recombination current are identical in both cases.

We conclude that the QFLS reduction deduced from the EL spectra modeling shown in Fig. 1 must also be the same under illuminated conditions at the same bias. This confirms the observation on the GaAs/InGaAs strained SQW [START_REF] Ekins-Daukes | [END_REF][START_REF] Ekins-Daukes | Proc. 3 rd World Conf. on Photovoltaic Energy Conversion[END_REF] and suggests the QWSC does not conform to the assumptions by Luque and Marti [START_REF] Luque | [END_REF] of a constant QFLS separation. Work is in progress examining the temperature of carriers in the wells and shows signs of higher carrier temperatures in the well mentioned above [8, 12,]. We are also extending these QFLS measurements to higher biases where radiative recombination dominates [START_REF] Barnham | Proc. 19th European Photovoltaic Solar Energy Conference and Exhibition[END_REF].

The EL spectra at various biases were integrated to calculate radiative currents shown as scattered square points in Fig. 3.a for the single well SB-QWSC sample. The measured total dark currents are shown as dots. The radiative recombination current is proportional to e qV/nkT , with an ideality factor n of 0.95±0.04, showing good agreement with the n = 1 expected for radiative processes. This last observation gives confidence into the procedure used to extract the radiative recombination current. The experimental radiative current was found to be significantly below the current predicted by the model assuming the QFLS was equal to the applied bias, as shown by the full line in Fig. 3.a, confirming that radiative recombination is reduced over the whole range of bias. The corresponding QFLS reduction for each applied bias is shown in Fig. 3.b and varies slightly between 15 and 19 meV depending on the applied bias. The radiative current becomes comparable with the total dark current at ~ 1.1V consistent with the change of ideality observed in SB-QWSCs at these biases which correspond to 200x concentration [START_REF] Barnham | Proc. 19th European Photovoltaic Solar Energy Conference and Exhibition[END_REF]. For terrestrial applications SB-QWSCs will be used under high light concentration to reduce costs. Hence a QFLS reduction at these biases is of great interest in the aim of minimizing dark currents and enhancing efficiencies. To exploit the QFLS observed in these low well number devices efficient forms of photon-recycling will be necessary for example by means of back surface mirrors as already under study [START_REF] Connolly | Proc. International Conference on Solar Concentrators for the Generation of Electricity or Hydrogen (ICSC-4)[END_REF].

In summary we have confirmed that the QFLS reduction first observed in SQW and DQW lattice matched and strained SQW cells is also observed in the higher material quality strain-balanced cells. We have presented evidence of a reduced radiative recombination current observed on three SB-QWSCs with 1 and 5 wells and interpreted as a reduction of the QFLS in the wells. This reduction could be explained by a temperature gradient in the nanostructures with "hot carriers" in the wells and would be a contribution to efficiency enhancement in low-well-number QWSCs. The QFLS reduction was observed in the dark with EL measurements as well as in the light with PL/EL measurements at V OC in contrast to some theoretical expectations. The reduction of the radiative losses was observed for several different biases for the single well SB-QWSC. The QFLS reduction was found to decrease with the number of wells which suggest that effective photon recycling will be necessary to fully exploit this effect. It will be important to extend these studies to higher biases and under higher light concentrations where the recombination is radiatively dominated and where hot carrier effects may be more important.

Figure captions

Fig. 1 Fig. 2

 12 Fig. 1Experimental (circles) and modeled EL (broken and full lines) of (a) the single

Fig. 3 : 2 E

 32 Fig. 3: (a) experimental radiative current (squares) and total dark current (dots)

Acknowledgments

Financial support is gratefully acknowledged from the European Marie Curie programme, contract No MEIF-CT-2003-501974.


Minjia Shi
email: mjshi@ahu.edu.cn

• Wang

Patrick Solé

Two families of two-weight codes over Z 4

Keywords: Two-weight codes, Projective codes, Strongly regular graphs Mathematics Subject Classification 94B 05, 05E 30

.

Introduction

An organic connection between two-weight codes over fields and strongly regular graphs (SRGs) was discovered in the 1970s [START_REF] Delsarte | Weights of linear codes and strongly regular normed spaces[END_REF], and is well-documented in [START_REF] Brouwer | Strongly regular graphs[END_REF]. A partial census of known constructions based on the arithmetic of finite fields appeared in the classic paper [START_REF] Calderbank | The geometry of two-weight codes[END_REF]. More recently, several new techniques of construction involve trace codes over different rings, and Gray maps [START_REF] Shi | Two new families of two-weight codes[END_REF][START_REF] Shi | Optimal two-weight codes from trace codes over F 2 + uF 2[END_REF][START_REF] Shi | Two and three weight codes over F p + uF p[END_REF][START_REF] Shi | On two-weight Z 2 k -codes[END_REF][START_REF] Shi | Optimal p-ary codes from one-weight codes and two-weight codes over F p + vF p[END_REF][START_REF] Shi | One-weight and two-weight Z 2 Z 2 [u, v]-additive codes[END_REF]. Especially, in [START_REF] Shi | Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over Z 4[END_REF], the authors focused on the construction of one-weight and two-weight codes over Z 4 . Later, in [START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF], the authors considered the linearity of the constructed codes. A natural question then, is whether there are other kinds of constructions of two-weight codes over Z 4 .

Motivated by the works listed above, in the present paper, we revisit two-weight codes over Z 4 and their linearity. This alphabet has been on the forefront of research in the domain of codes over rings since the prize awarded paper [START_REF] Hammons | The Z 4 -linearity of kerdock, preparata, goethals and related codes[END_REF]. More background material can be found in the recent book [START_REF] Shi | Codes and Rings: Theory and Practice[END_REF]. We give an infinite family of two-weight projective codes over Z 4 by their explicit generator matrices. Their Gray images are proved to be nonlinear with the same weight distribution as the two-weight binary codes of type SU1 from [START_REF] Calderbank | The geometry of two-weight codes[END_REF]. The coset graphs of the dual codes are shown to be SRGs, and determined completely. Thus they produce the Communicated by P. Charpin.

any vector x = (x 1 , x 2 , . . . , x n ) ∈ Z n 4 , we define its Lee weight as

W L (x) = n i=1 W L (x i ).
An N -Lee weight code is a code such that the cardinality of the set of nonzero Lee weights is N .

Moreover, A w i is the number of codewords of the nonzero Lee weight w i in C, where

1 ≤ i ≤ N .
The order of a nonzero codeword c ∈ C, is the smallest positive integer k such that kc = 0.

Each element x ∈ Z 4 has a 2-adic expansion x = α(x) + 2β(x), where α(x), β(x) ∈ F 2 .

The Gray map from Z 4 to F 2 2 is defined by (x) = (β(x), α(x) + β(x)). This map can be extended to Z n 4 naturally. is a weight-preserving map from (Z n 4 , Lee weight) to (F 2n 2 ,

Hamming weight), that is to say W L (x) = W H ( (x)).

If x = (x 1 , x 2 , . . . , x n ) and y = (y 1 , y 2 , . . . , y n ) are two arbitrary elements of Z n 4 , the inner product of x and y in Z n 4 is defined by x • y = x 1 y 1 + x 2 y 2 + • • • + x n y n , and the componentwise multiplication * of x and y is x * y = (x 1 y 1 , x 2 y 2 , . . . , x n y n ), where the operation is performed in Z 4 . The dual code of C is defined as

C ⊥ = {x ∈ Z n 4 |x • y = 0, ∀ y ∈ C}.
A Lee weight projective code C of length n over Z 4 is a linear code such that the minimum nonzero Lee weight of its dual code is at least three.

Graphs

In this note, all graphs are undirected, without loops or multiple edges. A graph is k-regular if all vertices have exactly k neighbors.

A k-regular graph on v vertices is a strongly regular graph (SRG) with parameters (v, k, λ, μ) if the number of common neighbors of any pair of vertices is λ or μ depending on the pair being connected or not.

The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A standard characterization of SRGs is that their spectrum consists of exactly three eigenvalues, namely k and two eigenvalues called the unrestricted eigenvalues [START_REF] Brouwer | Strongly regular graphs[END_REF]. 3 Preliminaries

If k = 1, let Y k = (1), when k ≥ 2, define Y k = ⎛ ⎝ Y k-1 Y k-1 Y k-1 Y k-1 B k-1 0 1 2 3 1 1 • • • 1 2 k-1 ⎞ ⎠ (1)
where B k-1 is a (k -1) × 2 k-1 matrix over 2Z 4 with all different columns and i is the row

vector (i, i, . . . , i) of length 2 k-2 (2 k-1 -1), i ∈ Z 4 .
Obviously, Y k is a k × n matrix over Z 4 where n = 2 k-1 (2 k -1). Let x i and x j be two columns of Y k , then it is easy to check that x i = mx j , where m = 1, 2, 3 and 1 ≤ i, j ≤ n. Let c = (x, 3x, y) ∈ C and the lengths of x and y are 2 k-1 (2 k -1) and 2 k -1, respectively.

= 2 k-1 (2 k -1), with the nonzero Lee weight w 1 = 2 k-1 (2 k -1), w 2 = 2 2k-1 . Moreover, A w 1 = 4 k -2 k and A w 2 = 2 k -1, respectively. Proof Let G k = Y k 3Y k B k , ( 2
)
then G k is a k × 4 k matrix over Z 4 . Since B k is a k × 2 k matrix
Obviously, W L (y) = 2 k or 0, and

W L (c) = W L (x) + W L (3x) + W L (y) = 2W L (x) + W L (y),
then, we have

W L (x) = W L (c) -W L (y) 2 = 4 k -W L (y) 2 . ( 3
) If W L (y) = 0, then W L (x) = w 1 = 2 k-1 (2 k -1). Otherwise, W L (x) = w 2 = 2 2k-1 . It is not difficult to get A w 2 = 2 k -1 when W L (y) = 0.
(r i ) = w 1 = 2 k-1 (2 k -1), but W L (2r i ) = w 2 = 2 2k-1 .
Step 1 Let

c = p 1 r 1 + p 2 r 2 + • • • + p k r k ,
where p i ∈ Z 4 , 1 ≤ i ≤ k and there exists p i = 0. We can prove that p i = 0 or 2 for all 1 ≤ i ≤ k, if the order of c is 2. By (1), let

r k = (0 1 2 3 2 k-1 1 1 . . . 1),
where i is the row vector (i, i, . . . , i) of length 2 k-2 (2 k-1 -1), i ∈ Z 4 . If the order of c is 2, and p k = 1 or 3, then c contains at least a 1 or a 3 according to the definition of Y k in (1), contradiction. Therefore, p k = 0 or 2 and p k r k = 0 or 2r k . So the order of [START_REF] Brouwer | Strongly regular graphs[END_REF], we can get p k-1 = 0 or 2. Then we can prove p i = 0 or 2 for all 1 ≤ i ≤ k.

p 1 r 1 + p 2 r 2 + • • • + p k-1 r k-1 is 2. Similarly, by
Step 2 By (3), we can get if

p i = 2 or 0, then W L ( p i r i ) = 2 2k-1 or 0, 1 ≤ i ≤ k.
Similarly,

W L ( p 1 r 1 + p 2 r 2 + • • • + p k r k ) = 2 2k-1 = w 2 ,
where p i = 0 or 2, 1 ≤ i ≤ k and there exists p i = 0. By Step 1, these are all the codewords of order 2. Therefore, if the order of c is 2, then W L (c) = 2 2k-1 = w 2 . Since A w 2 = 2 k -1 by Proposition 3.1 and C is a two-Lee weight code, we can get if the order of c is 4, then

W L (c) = 2 k-1 (2 k -1). When k ≥ 2, let X k = Y k-1 Y k-1 Y k-1 Y k-1 0 1 2 3 , ( 4
)
where i is the row vector

(i, i, . . . , i) of length 2 k-2 (2 k-1 -1), i ∈ Z 4 .
Similar to Proposition 3.1, we have

Proposition 3.3 If C is the code generated by X k , then C is a two-Lee weight code of length n = 2 k (2 k-1 -1) with the nonzero Lee weight w 1 = 2 k (2 k-1 -1), and w 2 = 2 2k-1 . Moreover, A w 1 = 4 k -2 k-1 and A w 2 = 2 k-1 -1, respectively.
Proof From the structure of X k and Proposition 3.1, we know there are 2 k-1 -1 codewords of weight 2 2k-1 generated by the first k -1 rows. Let r 1 , r 2 , . . . , r k be the rows of X k , and denote the codeword generated by the first k -1 rows by c. Without loss of generality, by (4), let

r k = (0 1 2 3),
where i is the row vector

(i, i, . . . , i) of length 2 k-2 (2 k-1 -1), i ∈ Z 4 . It is sufficient to prove W L (r k + c) = W L (2r k + c) = 2 k (2 k-1 -1). It is easy to check that W L (r k ) = W L (2r k ) = 2 k (2 k-1 -1). Let c = (c , c , c , c ), n i be the number of i in c
and n be the length of c . So n = 4n = 2 k (2 k-1 -1).

(1) If the order of c is 2, according to Corollary 3.2, W L (c) = 2 2k-1 , n = n 0 + n 2 , and (2) If the order of c is 4, n = n 0 + n 1 + n 2 + n 3 , and

W L (r k + c) = W L (c ) + W L (c + 1) + W L (c + 2) + W L (c + 3) = 2n 2 + n + 2(n -n 2 ) + n = 4n = 2 k (2 k-1 -1).
W L (r k + c) = W L (c ) + W L (c + 1) + W L (c + 2) + W L (c + 3) = (n 1 + 2n 2 + n 3 ) + (n 0 + 2n 1 + n 2 ) + (2n 0 + n 1 + n 3 ) + (n 0 + n 2 + 2n 3 ) = 4(n 1 + n 3 + n 2 + n 0 ) = 4n = 2 k (2 k-1 -1).
Similarly, we can also prove

W L (2r k +c) = 2 k (2 k-1 -1) = w 1 . Therefore, A w 2 = 2 k-1 -1,
and (

A w 1 = 4 k -2 k-1 . Example 3.4 If k = 3, then n = 24, w 1 = 24, w 2 = 32,
) 5
4 Two-lee weight projective codes over Z 4

In this section, we will give some structures of two-Lee weight projective codes over Z 4 .

Lemma 4.1 Let C be a linear code over

Z 4 with type 4 k 1 2 k 2 of length n. If k 2 = 1
and the columns of the generator matrix G (k 1 +1)×n are all distinct nonzero vectors

(c 1 , c 2 , . . . , c k 1 , c k 1 +1 ) T , where (c 1 , c 2 , . . . , c k 1 ) T is one column of Y k 1 , c k 1 +1 = 0 or 2,
then C is a two-Lee weight code of length n = 2 k 1 (2 k 1 -1) with the nonzero Lee weights

w 1 = 2 k 1 (2 k 1 -1) and w 2 = 2 2k 1 . Moreover, A w 1 = 2 k 1 (2 k 1 +1 -1) and A w 2 = 2 k 1 -1,

respectively.

Proof Without loss of generality, let

G (k 1 +1)×n = Y k 1 Y k 1 0 2 , ( 6
)
where i is the row vector

(i, i, . . . , i) of length 2 k 1 -1 (2 k 1 -1), i ∈ 2Z 4 and r 1 , r 2 , . . . , r k 1 , r k 1 +1
be the rows of G (k 1 +1)×n . Since the columns of G (k 1 +1)×n are all nonzero distinct vectors, it

follows that n = 2 k 1 (2 k 1 -1). Let c = (x, x) ∈ C, where x is generated by Y k 1 . According to Proposition 3.1, then W L (c) = 2W L (x) = 2 k 1 (2 k 1 -1) or 2 2k 1 .
Denote the number of i in x by n i , and the length of x by n . Obviously,

n = 2n = 2 k 1 (2 k 1 -1).
(1) If the order of x is 4, n = n 0 + n 1 + n 2 + n 3 , then we have

W L (r k 1 +1 + c) = W L (x) + W L (x + 2) = n 1 + 2n 2 + n 3 + 2n 0 + n 1 + n 3 = 2(n 0 + n 1 + n 2 + n 3 ) = 2n = 2 k 1 (2 k 1 -1).
(2) If the order of x is 2, n = n 0 + n 2 , n 1 = n 3 = 0, then we have

W L (r k 1 +1 + c) = W L (x) + W L (x + 2) = 2n 2 + 2n 0 = 2n = 2 k 1 (2 k 1 -1).
Hence, C is a two-Lee weight code with the nonzero Lee weights w 1 = 2 k 1 (2 k 1 -1), and

w 2 = 2 2k 1 . Obviously, A w 2 does not change comparing with Y k 1 , i.e. A w 2 = 2 k 1 -1, thus, A w 1 = 2 k 1 (2 k 1 +1 -1).
c k 1 +1 , . . . , c k 1 +k 2 ) T , where (c 1 , c 2 , . . . , c k 1 ) T is one column of Y k 1 , c i = 0 or 2, k 1 + 1 ≤ i ≤ k 1 + k 2 , then C is a two-Lee weight code of length n = 2 k 1 +k 2 -1 (2 k 1 -1) with the nonzero Lee weights w 1 = 2 k 1 +k 2 -1 (2 k 1 -1) and w 2 = 2 2k 1 +k 2 -1 . Moreover, A w 1 = 2 k 1 (2 k 1 +k 2 -1)
and

A w 2 = 2 k 1 -1, respectively.
Proof If k 2 = 1, the result follows from Lemma 4.1. Assume the result is valid for any k 2 = m, we denote the generator matrix and the nonzero weights by G k 1 ,m and by w 1 =

2 k 1 +m-1 (2 k 1 -1), w 2 = 2 2k 1 +m-1
, respectively, then for k 2 = m + 1, without loss of generality, the generator matrix G k 1 ,m+1 of C can be expressed as

G k 1 ,m+1 = G k 1 ,m G k 1 ,m 0 2 , ( 7
)
where i is the row vector (i, i, . . . , i) of length 2

k 1 +m-1 (2 k 1 -1), i ∈ 2Z 4 .
It's easy to verify that C is a two-Lee weight code with

w 1 = 2w 1 = 2 k 1 +m (2 k 1 -1) and w 2 = 2w 2 = 2 2k 1 +m .
Similar to the proof of Lemma 4.1, we can get

A w 2 = 2 k 1 -1 and A w 1 = 2 k 1 (2 k 1 +k 2 -1).
By induction hypothesis, the results are valid.

tors (c 1 , c 2 , . . . , c k 1 , c k 1 +1 , . . . , c k 1 +k 2 ) T , where (c 1 , c 2 , . . . , c k 1 ) T is one column of X k 1 defined above, c i = 0 or 2, k 1 + 1 ≤ i ≤ k 1 + k 2 , then C is a two-Lee weight code of length n = 2 k 1 +k 2 (2 k 1 -1 -1) with the nonzero Lee weights w 1 = 2 k 1 +k 2 (2 k 1 -1 -1), w 2 = 2 2k 1 +k 2 -1 . Moreover, A w 1 = 4 k 1 2 k 2 -2 k 1 -1 and A w 2 = 2 k 1 -1 -1, respectively.
Proof Assume that k 1 ≥ 2, then according to Theorem 4.2, we can construct a code C with type 4 k 1 -1 2 k 2 , and denote its generator matrix by G . By the definition of X k in (4), without loss of generality, let

G (k 1 +k 2 )×n = 0 1 2 3 G G G G , ( 10
)
where i is the row vector (i, i, . . . , i) of length 2

k 1 +k 2 -2 (2 k 1 -1 -1), i ∈ Z 4 .
It is easily seen that C is a two-Lee weight code with the nonzero weight w 1 = 2 k 1 +k 2 (2 k 1 -1 -1) and (1) every column of G contains 1 or 3;

w 2 = 2 2k
(2) any two columns of G are not multiple of each other by ±1. Case 1 Without loss of generality, let c = (0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0), where c i = c j = 1, i < j. Then x i + x j = 0. Thus, x j = 3x i , x i and x j are in proportion by the unit 3, contradiction.

Proof

Case 2 Without loss of generality, let c = (0, . . . , 0, 1, 0, . . . , 0, 3, 0, . . . , 0), where c i = 1, c j = 3, i < j. Then x i + x j = 0. Thus, x j = x i , contradiction.

Therefore, W L (c) ≥ 3. By definition, C is projective. Now we begin to prove the necessity. If (1) doesn't hold, let

x j ∈ 2Z k 1 +k 2 4
, then the codeword c = (0, . . . , 0, 2, 0, . . . , 0) ∈ C ⊥ , where c j = 2. If (2) doesn't hold, x i = ±x j for some pair of indices i < j.

Case 1 If x i = x j , we can construct c = (0, . . . , 0, 1, 0, . . . , 0, 3, 0, . . . , 0), where

c i = 1, c j = 3, i < j and c ∈ C ⊥ . Case 2 If x i = 3x j , then c = (0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0) ∈ C ⊥ , where c i = c j = 1.
But the Lee weight of c, c and c is 2. Contradict with w L (C ⊥ ) = 3. Step 1 Prove Y k 1 defined in ( 1) can make it. The result is obvious.

Step 2 Use induction for k 2 . If k 2 = 1, we know the generator matrix has the form projective. Assume the result is valid for any k 2 = m, then for k 2 = m + 1, according to [START_REF] Shi | Codes and Rings: Theory and Practice[END_REF] in Theorem 4.2, without loss of generality, the generator matrix

G k 1 ,m+1 of C is G k 1 ,m+1 = G k 1 ,m G k 1 ,m 0 2 , ( 13
)
where i is the row vector (i, i, . . . , i) of length 2 k 1 +m-1 (2 Similar to the proof of Theorem 4.9, we have the following corollary.

Corollary 4.10

The linear code obtained in Corollary 4.5 is projective.

Gray images

For x = (x 1 , x 2 , . . . , x n ) ∈ Z n 4 , define α(x) = (α(x 1 ), α(x 2 ), . . . , α(x n )), where α(0) = α(2) = 0 and α(1) = α(3) = 1.
Lemma 5.1 (See [START_REF] Hammons | The Z 4 -linearity of kerdock, preparata, goethals and related codes[END_REF]) Let C be a linear code over Z 4 , x 1 , x 2 , . . . , x m be a set of generators of

C, then (C) is linear if and only if 2α(x i ) * α(x j ) ∈ C for all i, j, satisfying 1 ≤ i ≤ j ≤ m.
Let q = p m , where p is prime and m is a positive integer. According to [START_REF] Calderbank | The geometry of two-weight codes[END_REF], an

[n, k] code C over G F(q) is a k-dimensional subspace of G F(q) n .
Next proposition is from the classical paper in [START_REF] Calderbank | The geometry of two-weight codes[END_REF] about the survey of two weight codes over finite fields.

Proposition 5.2 (See [START_REF] Calderbank | The geometry of two-weight codes[END_REF]) Example SU 1: Let be the complement of a t-dimensional subspace of G F(q) k where 1 ≤ t ≤ k -1. If v is a non-zero vector then |v ⊥ ∩ | = q k-1 -q t or q k-1q t-1 . For a two-weight code with Example SU1, if the field is G F(q), its parameters are (see FIGURE 1a and FIGURE 2a in [START_REF] Calderbank | The geometry of two-weight codes[END_REF])

n = q l -q t q -1 , k = l, w 1 = q l-1 -q t-1 , w 2 = q l-1 , A w 1 = q l -q l-t , A w 2 = q l-t -1.
where 1 ≤ t ≤ l -1.

Proposition 5.3

The Gray images of the codes of Theorem 4.2 and Corollary 4.5 are binary distance invariant codes with the weights given w 1 = q l-1q t-1 and w 2 = q l-1 .

Note that we do not claim these binary codes to be linear.

Here is a table about the parameters of the Gray images (C 1 ), (C 2 ) and (C 3 ), where Table 1 The parameters of (C 1 ), (C 2 ) and (C 3 )

n w 1 w 2 A w 1 A w 2 SU 1
q l -q t q-1 q l-1q t-1 q l-1 q lq l-t q l-t -1

(C 1 ) 2 k 1 +k 2 (2 k 1 -1) 2 k 1 +k 2 -1 (2 k 1 -1) 2 2k 1 +k 2 -1 4 k 1 2 k 2 -2 k 1 2 k 1 -1 (C 2 ) 2 k 1 +k 2 (2 k 1 -2) 2 k 1 +k 2 (2 k 1 -1 -1) 2 2k 1 +k 2 -1 4 k 1 2 k 2 -2 k 1 -1 2 k 1 -1 -1 (C 3 ) 2 2k 1 +k 2 -1 2 2k 1 +k 2 -2 2 2k 1 +k 2 -1 4 k 1 2 k 2 -2 1
Finally, we will consider the linearity of the image codes in Theorem 4. Proof When k 1 = 1, the result can be obtained in Corollary 6.8 in [START_REF] Shi | Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over Z 4[END_REF]. When k 1 ≥ 2, denote the generator matrix of C by G k 1 ,k 2 and let r 1 , r 2 , . . ., r k 1 , . . ., r k 1 +k 2 be the rows of

G k 1 ,k 2 ,
where r j is a vector over 2Z 4 ,

k 1 + 1 ≤ j ≤ k 1 + k 2 . By Theorem 4.2, G k 1 +1,k 2 = G k 1 ,k 2 G k 1 ,k 2 0 2 , ( 14
)
where i is the row vector (i, i, . . . , i) of length 2

k 1 +k 2 -1 (2 k 1 -1), i ∈ 2Z 4 . Obviously, if
either r 1 or r 2 is from the last k 2 rows, it's easy to check that 2α(r 1 ) * α(r 2 ) = 0 ∈ C.

(

) If k 1 = k + 1 ≥ 2, k 2 = 0, then r 1 , 1
Y k+1 = Y k Y k Y k Y k B k 0 1 2 3 1
where k ≥ 1, B k is a k × 2 k matrix over 2Z 4 with all different columns, i is a row vector (i, i, . . . , i) of length 2 k-1 (2 k -1), i ∈ Z 4 and 1 is the row vector (1, 1, . . . , 1) of length

2 k .
Without loss of generality, let

r 1 = ( 2 k-1 (2 k -1) 0 0 . . . 0 2 k-1 (2 k -1) 1 1 . . . 1 2 k-1 (2 k -1) 2 2 . . . 2 2 k-1 (2 k -1) 3 3 . . . 3 2 k 1 1 . . . 1) (15)
be the last row of Y k+1 , and (2) If k 2 ≥ 1, then the first k 1 rows of G k 1 ,k 2 can be written as:

r 2 = (x
Y k 1 Y k 1 . . . Y k 1 2 k 2
, and let

r i = (r i r i . . . r i ) 2 k 2
, where i = 1 or 2. Since k 1 = k + 1, we have

W L (2α(r 1 ) * α(r 2 )) = 2 k 2 W L (2α(r 1 ) * α(r 2 )) = 2 k 2 +2 W L (α(x)) ≤ 2 k 2 +2 × 2 k-1 (2 k -1) = 2 k 1 +k 2 (2 k 1 -1 -1).
By Theorem 4.2, C is a two-Lee weight code with the nonzero Lee weights 2

k 1 +k 2 -1 (2 k 1 -1)
and 2 2k 1 +k 2 -1 . And the nonzero smallest Lee weight is 2

k 1 +k 2 -1 (2 k 1 -1) > 2 k 1 +k 2 (2 k 1 -1 -1).
+ 1 ≤ j ≤ k 1 + k 2 .
(1) By Corollary 4.5, if k 1 = 2, k 2 = 1, the generator matrix is:

G 2,1 = ⎛ ⎝ 1111 1111 0123 0123 2222 0000 ⎞ ⎠ .
So, if r 1 , r 2 , r j are the rows of G 2,k 2 , 3 ≤ j ≤ k 2 + 2, we can get r 1 is all 1 vector. Therefore,

α(r 1 ) = 1 and α(r j ) = 0. Besides, 2α(r 1 ) * α(r 2 ) = 2r 2 ∈ C, 2α(r 1 ) * α(r j ) = 0 ∈ C, 2α(r 2 ) * α(r j ) = 0 ∈ C, 2α(r j ) * α(r k ) = 0 ∈ C, where 3 ≤ j, k ≤ 2 + k 2 . By Lemma 5.1, (C) is linear when k 1 = 2, k 2 ≥ 0.
(2) If k 1 ≥ 3, by [START_REF] Shi | Optimal two-weight codes from trace codes over F 2 + uF 2[END_REF] in the proof of Corollary 4.5, where G k 1 -1,k 2 is the generator matrix of the code C obtained in Theorem 4.2, and i is the row vector (i, i, . . . , i) of length 2 k 1 +k 2 -2 (2 k 1 -1 -1), i ∈ Z 4 . Without loss of generality, let r 1 = (0 1 2 3) and r 2 = (r 0 r 0 r 0 r 0 )

G k 1 ,k 2 = 0 1 2 3 G k 1 -1,k 2 G k 1 -1,k 2 G k 1 -1,k 2 G k 1 -1,k 2 ,
where r 0 is one row of G k 1 -1,k 2 . Then 2α(r 1 ) * α(r 2 ) = 2(0 α(r 0 ) 0 α(r 0 )).

Since the length of r 0 is 2

k 1 +k 2 -2 (2 k 1 -1 -1), W L (2α(r 1 ) * α(r 2 )) = 4W L (α(r 0 )) ≤ 2 k 1 +k 2 (2 k 1 -1 -1). ( 16
) If 0 = 2α(r 1 ) * α(r 2 ) ∈ C, since C is a two-Lee weight code by Corollary 4.5, W L (2α(r 1 ) * α(r 2 )) = 2 k 1 +k 2 (2 k 1 -1 -1) or 2 2k 1 +k 2 -1 . Thus 2 k 1 +k 2 (2 k 1 -1 -1) ≤ W L (2α(r 1 ) * α(r 2 )) = 4W L (α(r 0 )). ( 17
)
Therefore, combining ( 16) and (17), we have

W L (2α(r 1 ) * α(r 2 )) = 4W L (α(r 0 )) = 2 k 1 +k 2 (2 k 1 -1 -1),
which forces α(r 0 ) to be all 1 vector. Thus we have

W L (r 0 ) = 2 k 1 +k 2 -2 (2 k 1 -1 -1), W L (2r 0 ) = 2 k 1 +k 2 -1 (2 k 1 -1 -1)
and then W L (2r 0 ) = 2W L (r 0 ). However, r 0 is one row of

G k 1 -1,

Graphs

We recall the classic lemma from [START_REF] Shi | On two-weight Z 2 k -codes[END_REF]. From this classic lemma, and the characterization of SRGs given in §2.2, the following two results follow immediately. Their proofs are omitted.

Remarks:

• Following [1, §1.1.3] we see that these graphs are complete multipartite of type K a×m with m = 2 k 1 +k 2 for Theorem 4.2 or m = 2 k 1 +k 2 -1 for Corollary 4.5, and, in both cases, a = |C| m .

• The graph corresponding to the code of Example 4.6 is bipartite complete, since the code is an Hadamard code of a kind studied in [START_REF] Krotov | Z 4 -linear perfect codes[END_REF][START_REF] Krotov | Z 4 -Linear Hadamard and Extended Perfect Codes, WCC2001[END_REF].

Conclusion

In this paper, we have constructed two infinite families of projective two-weight Z 4 -codes.

Further, their Gray images are nonlinear, and have the same weight distributions as a family of two-weight codes of [START_REF] Calderbank | The geometry of two-weight codes[END_REF], the SU1 family. In small values of parameters these codes can be implemented in the computer package Magma. It is a worthwhile project to check which families of binary two-weight codes from [START_REF] Calderbank | The geometry of two-weight codes[END_REF] admit Z 4 -linear analogues.

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Corollary 3 . 2

 32 Let C be the code generated by Y k , c ∈ C and c = 0, then W L (c) = 2 2k-1 if the or der o f c is 2, 2 k-1 (2 k -1) if the or der o f c is 4. R e v i s e d P r o o f M. Shi et al. Proof Let r 1 , r 2 , . . ., r k be the rows of Y k . From the proof of Proposition 3.1, W L

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  then, according to Proposition 3.3, there is a two-Lee weight code with the generator matrix:

123

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Example 4 . 3 Example 4 . 4 Corollary 4 . 5

 434445 If k 1 = 2, k 2 = 1, then n = 12, w 1 = 12, w 2 = 16, then, according to Theorem 4.2, there is a two-Lee weight code with the generator matrix: If k 1 = 2, k 2 = 2, then n = 24, w 1 = 24, w 2 = 32, then, according to Theorem 4.2, there is a two-Lee weight code with the generator matrix: Let C be a linear code over Z 4 with type 4 k 1 2 k 2 of length n. When k 1 ≥ 2, the columns of the generator matrix G (k 1 +k 2 )×n are all distinct nonzero vec-

Let x 1 ,

 1 x 2 , . . . , x n be the columns of G. Let us first prove the sufficiency. Let C ⊥ be the dual code of C. If c ∈ C ⊥ , then we only need to prove W L (c) ≥ 3. Obviously, W L (c) cannot be 1 because all columns are nonzero. Assume W L (c) = 2. Obviously, c cannot be written as c = (0, . . . , 0, 2, 0, . . . , 0) due to (1). Then we have 2 cases:

Theorem 4 . 9

 49 Let C be a linear code over Z 4 with type 4 k 1 2 k 2 of length n obtained in Theorem 4.2, then C is projective. Proof Denote the generator matrix of C by G k 1 ,k 2 , then it is sufficient to prove that G k 1 ,k 2 satisfies Lemma 4.8.

( 6 )

 6 in Lemma 4.1. And by Step 1 and Lemma 4.8, we can easily prove that the code is 123 Journal: 10623 Article No.: 0796 TYPESET DISK LE CP Disp.:2020/9/3 Pages: 13 Layout: Small R e v i s e d P r o o f M. Shi et al.

C 1 andRemark 5 . 4 Remark 5 . 5

 15455 C 2 are the codes obtained in Theorem 4.2 and Corollary 4.5 in this paper, respectively, and C 3 is the code obtained in Theorem 5.6 in [14]. If k 1 = 1, (C 1 ) and (C 3 ) have the same parameters. Besides, due to Corollary 6.8 in [14], when k 1 = 1, (C 1 ) reaches the Plotkin bound and Griesmer bound. It can be seen that the values of l for (C 1 ), (C 2 ) and (C 3 ) are all 2k 1 + k 2 . But the values of t are k 1 + k 2 , k 1 + k 2 + 1, 2k 1 + k 2 -1 for (C 1 ), (C 2 ) and (C 3 ), respectively. Therefore, these three families of codes lead to non-equivalent two-Lee weight linear codes. 123 Journal: 10623 Article No.: 0796 TYPESET DISK LE CP Disp.:2020/9/3 Pages: 13 Layout: Small R e v i s e d P r o o f Two families of two-weight codes over Z 4

2 and Corollary 4. 5 . 5 . 6

 556 Theorem If the columns of the generator matrix of C satisfy the conditions of Theorem 4.2, then (C) is linear when k 1 = 1 and nonlinear when k 1 ≥ 2.

x x x b), where x is

  a row of Y k , and b is a row of B k . Then2α(r 1 ) * α(r 2 ) = 2(α(0) α(1) α(2) α(3) α(1 ) * (α(x) α(x) α(x) α(x) α(b)) = 2(0 α(x) 0 α(x) 0), we get W L (2α(r 1 ) * α(r 2 )) = 4W L (α(x)). Since the length of x is 2 k-1 (2 k -1), then 4W L (α(x)) ≤ 4 × 2 k-1 (2 k -1) = 2 k+1 (2 k -1).123 Journal: 10623 Article No.: 0796 TYPESET DISK LE CP Disp.:2020/9/3 Pages: 13 Layout: Small R e v i s e d P r o o f M. Shi et al. By Proposition 3.1, C is a two-Lee weight linear code with the nonzero Lee weights 2 k (2 k+1 -1), and 2 2k+1 . And the nonzero smallest Lee weight of C is 2 k (2 k+1 -1) > 2 k+1 (2 k -1). Thus, 2α(r 1 ) * α(r 2 ) / ∈ C. By Lemma 5.1, (C) is nonlinear when k 1 ≥ 2, k 2 = 0.

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k 2 ,Example 5 . 9

 259 the generator matrix obtained in Theorem 4.2, and the two Lee weights w 1 and w 2 can not satisfy w 2 = 2w 1 , contradiction. Thus there are at least 2 rows r 1 , r 2 satisfying 2α(r 1 ) * α(r 2 ) / ∈ C. Namely, (C) is nonlinear when k 1 ≥ 3, k 2 ≥ 0. Consider the code defined in Example 4.7. Let r 1 annd r 2 denote the first and second rows of G 3,1 in (12), respectively. Then the Lee weight of 2α(r 1 ) * α(r 2 ) = 2α(111102111102111102111102111102111102111102111102) * α(012311012311012311012311012311012311012311012311) = (020200020200020200020200020200020200020200020200) is 32 which can not be obtained in C since C is a two-Lee weight code with w 1 = 48, w 2 = 64.

Lemma 6 . 1

 61 If C is a Z 4 -code of length n and weights w 1 , w 2 the coset graph of C ⊥ is an SRG on |C| vertices, degree 2n, unrestricted eigenvalues 2n -2w 1 and 2n -2w 2 .

Theorem 6 . 2 ( 1 )

 621 If C is the code of Theorem 4.2 then the coset graph of C ⊥ is an SRG on |C| vertices, degree 2n, and unrestricted eigenvalues 0 and -2 k 1 +k 2 . (2) If C is the code of Corollary 4.5 then the coset graph of C ⊥ is an SRG on |C| vertices, degree 2n, and unrestricted eigenvalues 0 and -2 k 1 +k 2 +1 . 123 Journal: 10623 Article No.: 0796 TYPESET DISK LE CP Disp.:2020/9/3 Pages: 13 Layout: Small R e v i s e d P r o o f M. Shi et al.

  Besides, if we delete 0 ( a column of B k ) from B k , then we get a k × (2 k -1) matrix B k over 2Z 4 . Let C be the code generated by B k . It's easy to check that C is a special case of the code obtained in Theorem 4.5 in[START_REF] Shi | Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over Z 4[END_REF], when k 1 = 0, k 2 = k, then C is a one-Lee weight linear code with the nonzero Lee weight w = 2 k .

The following proposition is an alternative proof of Property 5.68 in (Handbook of Coding Theory, North Holland, 1998, WC Huffman, V. Pless eds, p. 1561).

Proposition 3.1 If C is the code generated by Y k , then C is a two-Lee weight code of length n

  Theorem 4.2 Let C be a linear code over Z 4 with type 4 k 1 2 k 2 of length n. If the columns of the generator matrix G (k 1 +k 2 )×n are all distinct nonzero vectors (c 1 , c 2 , . . . , c k 1 ,

  1

  Two families of two-weight codes over Z 4 If k 1 = 2, k 2 = 2, then n = 16, w 1 = 32, w 2 = 16, according to Corollary 4.5, there is a two-Lee weight code with the generator matrix: If k 1 = 3, k 2 = 1, then n = 48, w 1 = 48, w 2 = 64, according to Corollary 4.5,

	Example 4.6 G 2,2 =	⎛ ⎜ ⎜ ⎝	1111 1111 1111 1111 0123 0123 0123 0123 0000 0000 2222 2222	⎞ ⎟ ⎟ ⎠ .	(11)
				0000 2222 0000 2222
	Example 4.7 there is a two-Lee weight code with the generator matrix G 3,1 :
	⎛	111102 111102 111102 111102 111102 111102 111102 111102	⎞
	⎜ ⎜ ⎝	012311 012311 012311 012311 012311 012311 012311 012311 000000 111111 222222 333333 000000 111111 222222 333333	⎟ ⎟ ⎠ .	(12)
		222222 222222 222222 222222 000000 000000 000000 000000

+k 2 -1 . By induction hypothesis, C is a two-Lee weight code over Z 4 . 123 Journal: 10623 Article No.: 0796 TYPESET DISK LE CP Disp.:2020/9/3 Pages: 13 Layout: Small R e v i s e d P r o o f

Next, we are ready to give the sufficient and necessary conditions for a two-Lee weight linear code to be projective over Z 4 . Lemma 4.8 Let C be a linear code over Z 4 with type 4 k 1 2 k 2 of length n and with the generator matrix G. Then C is projective if and only if the following conditions hold:

  k 1 -1), i ∈ 2Z 4 . So, all the columns of G k 1 ,m+1 contain a 1 or a 3. If there are two columns in proportion by ±1, then there are two columns of G k 1 ,m in proportion by ±1, a contradiction. So the result is valid for k 2 = m + 1. By induction hypothesis, G k 1 ,k 2 satisfies Lemma 4.8 for any k 1 , k 2 ≥ 1, then C is projective by Lemma 4.8.

  r 2 , . . ., r k and r k+1 become the rows of Y k+1 . By the definition of Y k , we can get

  Thus, 2α(r 1 ) * α(r 2 ) / ∈ C. By Lemma 5.1, (C) is nonlinear when k 1 ≥ 2, k 2 ≥ 0. Consider the code defined in Example 4.3. Let r 1 and r 2 denote the first and second rows of G 2,1 in (8), respectively. Then the Lee weight of

	Example 5.7

2α(r 1 ) * α(r 2 ) = 2α(111102111102) * α(012311012311) = 2(111100111100) * (010111010111) = (020200020200) is 8 which can not be obtained in C since C is a two-Lee weight code with w 1 = 12, w 2 = 16. Theorem 5.8 If the columns of the generator matrix of C satisfy the conditions of Corollary 4.5, then (C) is linear when k 1 = 2 and nonlinear when k 1 ≥ 3.

Proof Denote the generator matrix of C by G k 1 ,k 2 . Let r 1 , r 2 , . . . , r k 1 , . . . , r k 1 +k 2 be the rows of G k 1 ,k 2 , where r j is a vector over 2Z 4 , k 1

Acknowledgements

The authors thank Denis Krotov for helpful discussions. This research is supported by the National Natural Science Foundation of China (61672036), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), the Academic Fund for Outstanding Talents in Universities (gxbjZD03).

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