Samet Gelincik

Philippe Mary

Jean-Yves Baudais

Anne Savard

Achieving PAC Code Performance with SCL Decoding without Extra Computational Complexity

Keywords: Polar Codes, dynamic frozen function, Reed-Muller codes, successive cancellation list decoding

For finite blocklength polar codes, the minimum distance and the number of low weight codewords are essential to obtain good performance under successive cancellation list decoding with moderate and high list sizes. In this paper, we propose a code design method to decrease the number of low weight codewords for some information lengths with a very low computational complexity. In the proposed method, some information bits are encoded by several rows of the polar encoding matrix, i.e., each of the dynamic frozen bits is chosen the same as one of the preceding information bits. The dynamic frozen bit index set is determined by using the connection between the binary representation of the row indices and the number of common 1-bit positions of any given rows. The resulting design is shown to perform as well as polarization-adjusted-convolutional codes [9] under successive cancellation list decoding but with significant computational complexity savings. These findings pave the way for the use of polar codes in applications with stringent complexity and with low energy consumption constraints.

I. INTRODUCTION

Polar codes are the first proven (symmetric) capacityachieving block codes in asymptotic regime with low decoding complexity for discrete binary input memoryless channels [START_REF] Arikan | Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels[END_REF]. Polar codes have recently been adopted for control channels in 5G networks and may be used for other purposes in future wireless networks. Indeed in beyond 5G networks, use cases such as traffic safety, industrial control, medical and internet services will rely on highly reliable low latency communications [START_REF]Service requirements for the 5G system; Stage 1 (Release 17)[END_REF], [START_REF] Itu-T | Representative use cases and key network requirements for network 2030[END_REF]. Therefore, it is important to design polar codes that are efficient in short blocklength regime.

However, for short and moderate code sizes, polar codes under successive cancellation decoding performs poorly due to the incomplete polarization of the channel, and hence, designing codes with good minimum distance properties becomes very important to improve performance. For instance in [START_REF] Tal | List Decoding of Polar Codes[END_REF], cyclic redundancy check (CRC) concatenation is proposed to improve the minimum distance properties of the code to help successive cancellation list (SCL) decoding. In [START_REF] Zhang | CRC Code Design for List Decoding of Polar Codes[END_REF] and [START_REF] Piao | Approaching the Normal Approximation of the Finite Blocklength Capacity Within 0.025 dB by Short Polar Codes[END_REF], the CRC polynomial has been optimized to increase minimum Hamming distance and minimizing the number of low weight codewords by an exhaustive search over a wide range of CRC polynomials. This work has been partially supported by IRCICA, CNRS USR 3380, Lille, France and the French National Agency for Research (ANR) under grant ANR-16-CE25-0001 ARBurst.

Authors in [START_REF] Trifonov | Polar codes with dynamic frozen symbols and their decoding by directed search[END_REF] proposed to improve the minimum distance properties of a polar code by choosing the frozen bits as a function of preceding information bits. This technique has been called dynamic freezing and has been extended in [START_REF] Miloslavskaya | Design of Short Polar Codes for SCL Decoding[END_REF] by introducing the concept of precoded polar codes, in which any linear codes can be represented with a precoding matrix followed by a standard polar transformation matrix. In precoded polar codes, information bit indices are not necessarily chosen as it is in the original polar codes. In this article, we use the term polar-like codes to represent a general class of linear precoded codes with information and frozen bits 1 .

Recently, polarization-adjusted convolutional (PAC) code has been proposed by Arikan [START_REF] Arikan | From sequential decoding to channel polarization and back again[END_REF] and studied further in [START_REF] Arıkan | Systematic Encoding and Shortening of PAC Codes[END_REF] and [START_REF] Thibaud | On Systematic Polarization-Adjusted Convolutional (PAC) Codes[END_REF]. The information indices are chosen according to the Reed-Muller (RM) design rule, i.e., the rows of the encoding matrix with the highest Hamming weights are selected as information rows. PAC codes have been shown to perform close to the normal approximation (NA) of the second order rate [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF]. This method allows to reduce drastically the number of minimum weight codewords by employing an outer convolutional encoder that can be represented with an upper triangular matrix that in turn has the property to reduce the number of minimum weight codewords [START_REF] Li | On Pre-transformed Polar Codes[END_REF]. However, the improved performance comes at the cost of extra computational complexity. For instance, the decoding complexity of PAC codes is of the order of magnitude of SCL decoder's itself for short blocklengths. In [START_REF] Piao | Approaching the Normal Approximation of the Finite Blocklength Capacity Within 0.025 dB by Short Polar Codes[END_REF], state of the art has been moved a step forward by combining variable list size SCL decoding with a sphere decoder. The technique has been shown to approach the achievable error probability in finite block length but at the cost of very high complexity of the sphere decoder that may prevent from its usage in applications with low computational capability.

In this paper, we propose an alternative way for designing polar-like codes in the sense that some frozen bits are determined with a dynamic freezing function with a single argument, and not with a combination of preceding information bits, as it has been done in literature. The proposed process of determining the dynamic freezing functions is named row merging. The proposal is based on the relationship between the binary representation of row indices of the polar encoding matrix and the number of common 1-bit positions among the rows. In order to construct the code, the information bit indices are first chosen according to the Reed-Muller rule, which provides the highest minimum distance among the codewords without dynamic frozen bits. Then, we search for the rows that can be merged together in order to decrease the number of low weight codewords. Our method provides codes that perform as well as the PAC codes [START_REF] Arikan | From sequential decoding to channel polarization and back again[END_REF] in terms of frame error rate (FER) in binary-input additive white Gaussian noise (BI-AWGN) channel but at a much smaller computational complexity.

II. PRELIMINARIES A. Notations

Vectors of length N are represented in row and denoted with sans serif font x. The j-th entry of the vector x is x j with j ∈ N = {0, 1, • • • , N -1}. The set of integers from j to k -1 are represented by [j, k) and the set of integers from j to k are represented by [j, k]. A sub-vector drawn from x is denoted as

x k j = [x j , • • • , x k ], ∀(j, k) ∈ N 2
, j < k, and simply as x k if j = 0. Index sets are denoted by caligraphic letters, e.g. A, and are sorted in the ascending order. For a vector x and set A ⊂ N , x A = {x j : j ∈ A}. The matrices are denoted with uppercase sans serif font, e.g., G. For a matrix G ∈ {0, 1} N ×N and index set A ⊆ [0, N -1], G A denotes the matrix consisting of rows of G indexed by A. Modulo-2 binary addition is denoted by ⊕.

For any index j ∈ N , N = 2 n , the n-bit binary representation of j is denoted by b j and is a vector of length n. The indexing of elements in b j is started from the least significant bit, i.e., the rightmost bit, and b j,k is the k-th least significant bit, k ∈ [0, n -1]. The number of ones and zeros in any binary vector x is denoted by i 1 (x) and i 0 (x), respectively. For any x 1 , x 2 ∈ {0, 1} N , x 1 ∩x 2 is the element-wise logical AND operation and x 1 ∪x 2 is the element-wise logical OR operation. Moreover, P 1 (x), P 0 (x) denotes the index sets of 1's and 0's, respectively, of any binary vector x.

B. Polar-like and Polar Codes

Let C(N = 2 n , k, A), n ∈ N, be a polar-like code defined as a binary linear block code with generator matrix G N ,

G N = g T 0 g T 1 • • • g T N -1 T = G ⊗n Ker , G Ker = 1 0 1 1
where ⊗ is the Kronecker product and [•] T the transpose operator. The input bits are denoted by u ∈ {0, 1} N , where k of them are information bits and indexed by A ⊆ N , and the rest are indexed by F = N \ A and called frozen bits since they can be set to any binary value and are known a priori by the decoder. Alternatively, the jth row g j of generator matrix G N is represented as

g j = ĝb j,(n-1) ⊗ ĝb j,(n-2) ⊗ • • • ⊗ ĝbj,0 (1)
where ĝ0 = [1 0] and ĝ1 = [1 1].

Let W : X → Y be a binary-input symmetric discrete memoryless channel, where X = {0, 1}, with a priori probabilities p X (0) = p X (1) = 1/2, and W (y|x) are transition probabilities ∀(x, y) ∈ X × Y. It has been shown in [START_REF] Arikan | Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels[END_REF] that the application of a linear mapping such as x = uG N for any u ∈ X N creates synthetic subchannels W (j)

N : X → Y N × X j such as W (j) N (y, u j-1 |u j ) = u N -1 j+1 1 2 N -1 W N (y|uG N ) (2)
in which p(u) = 1/2 N and W N : X N → Y N denotes the vector channel which, in memoryless channel, becomes

W N (y|x) = N -1 j=0 W (y j |x j ). (3)
The polar code determines the information bit index set A as the k most reliable bits and the rest of the indices are the frozen bit set F and transmitted as u j = 0, ∀j ∈ F. To exploit the polarization phenomenon imposed by the linear transformation G N , the SC decoding has been proposed [START_REF] Arikan | Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels[END_REF]:

ûj = 0 if W (j) N (y, ûj-1 |0) ≥ W (j) N (y, ûj-1 |1) 1 otherwise, , j ∈ A (4)
where ûj = 0, ∀j ∈ F.

The dynamic frozen bits concept for polar codes has been first introduced in [START_REF] Trifonov | Polar codes with dynamic frozen symbols and their decoding by directed search[END_REF]. Here, we generalize the definition of [START_REF] Trifonov | Polar codes with dynamic frozen symbols and their decoding by directed search[END_REF] for polar-like codes and name them as dynamic polarlike codes, which is the same as precoded polar codes of [START_REF] Miloslavskaya | Design of Short Polar Codes for SCL Decoding[END_REF] and [START_REF] Miloslavskaya | Recursive Design of Precoded Polar Codes for SCL Decoding[END_REF]. In these codes, some frozen bits j ∈ F are determined dynamically using previous bits2 with a Boolean function h j : {0, 1} j → {0, 1}:

u j = h j (u 0 , u 1 , • • • , u j-1 ), j ∈ F (5)
For dynamic polar-like codes, the encoding process is:

x = m∈A u m • g m ⊕ j∈F h j • g j , (6)
where h j is a short-hand of h j (u 0 , u 1 , • • • , u j-1 ) and h j (•) = 0 if the jth bit is not a dynamic frozen bit.

The SC decoding of dynamic polar-like codes is the same as polar codes for information bits. However, the dynamic frozen bits are obtained with their corresponding functions

ûj = h j (û 0 , û1 , . . . , ûj-1 ) (7)
during the course of decoding. The reader may refer to [START_REF] Tal | List Decoding of Polar Codes[END_REF] for SCL decoding details.

C. Distance Spectrum and Symbol Error Probability

For a C(N, k) binary linear block code, the minimum distance d min is the minimum Hamming distance between any two codewords x, x ∈ C, that is to say

min x =x x,x ∈C i 1 (x ⊕ x ) = min x =0,x∈C i 1 (x) (8)
Let A w (C) be the number of codewords in C with the Hamming weight w, i.e.,

A w (C) := |{x : i 1 (x) = w, x ∈ C}|, (9)
Then, the distance properties of the code C is described by the set

A C = {A 0 , A 1 , • • • , A N }, which is called distance spectrum (or the weight enumerator function).
The decoding error probability of any code C(N, k) under maximum likelihood (ML) decoding, P ML , is upper bounded by the union bound. For the binary phase-shift keying modulation over AWGN channel, the union bound is given as [START_REF] Ma | New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes[END_REF]:

P ML ≤ P ub = N w=1 A w Q( √ w • SNR). ( 10
)
For high SNR, the union bound is well approximated by [START_REF] Zhang | CRC Code Design for List Decoding of Polar Codes[END_REF]

P ub ≈ A dmin Q( d min • SNR), (11)
which indicates that, at high SNR's, the minimum weight codewords contribute more to the union bound than codewords with higher Hammming weights. To estimate the distance spectrum, we adopt the SCL decoding method proposed in [START_REF] Li | An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check[END_REF]. For any polar-like code with u F = 0, the minimum distance is given as d min = 2 min j∈A i1(bj ) in [START_REF] Korada | Polar codes for channel and source coding[END_REF].

III. NEW CODE DESIGN WITH DYNAMIC FROZEN BITS

This section summarizes our main contribution in this paper.

A. Number of Common 1-bit Positions in G N Rows

The jth row of G N , g j , in (1) can be divided into n disjoint vectors r j ( ), such that for ∈ [1, n -1]:

r j ( ) = 0 2 -1 0 if b j, = 0 [r j (0)r j (1) • • • r j ( -1)] if b j, = 1 (12)
and r j (0) = ĝbj,0 . The length of r j ( ) is 2 if ≥ 1 and is equal to 2 if = 0. A close look at the recursive nature of r j ( ) reveals that the entries of g j , indexed by the set K , defined as

K = {k : b k, = 1, k ∈ N }, (13)
are zeros if b j, = 0, ∈ [0, n -1]. More precisely, the set K gets the following particular form

K = {k : k = a 2 +1 -b, a ∈ [1, 2 n-1-], b ∈ [0, 2 -1]}, (14)
and g j,K = 0 if b j, = 0 whatever b j, , = , is. We define the complementary set as K c = N \ K . Note that, by [START_REF] Li | On Pre-transformed Polar Codes[END_REF],

|K | = |K c | = 2 n-1 for any ∈ [0, n -1] since there are n -1 free positions in b k if the th bit is set to 1. Moreover | ∈B K |=| ∈B K c |= N/2 |B| ( 15
)
for any subset B ⊆ [0, n).

Theorem 1. Let G N be the generator matrix for a polar-like code (N = 2 n , k) with arbitrary information bit indices set A and frozen bit indices set F. For any T ⊆ N , the number of common 1-bit positions for all j ∈ T is given as:

i 1 j∈T g j = 2 i1( j∈T bj ) . (16)
Proof. By [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF] note that if b j, = 0, all entries of g j , i.e., g j,k , indexed by k ∈ K are zero. Therefore, the number of zeros at row vector g j is given by

i 0 (g j ) = n-1 =0, b j, =0 K (17)
and the number of ones is

i 1 (g j ) = N -i 0 g j (a) = | N \ n-1 =0, b j, =0 K | (b) = | n-1 =0, b j, =0 K c | (c) = 2 n 2 i0(bj ) = 2 i1(bj ) (18)
where (a) and (b) are due to standard operations on sets and (c) is due to [START_REF] Ma | New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes[END_REF] since |B| = i 0 (b j ) in (b). In general, for any T ⊆ N , the number of common ones is given as

i 1 ¯ j∈T g j = N -i 0 ¯ j∈T g j = | N \ : b j, =0, j∈T K | = | : b j, =0, j∈T K c | (a) = 2 n 2 i0( ∪j∈T bj ) = 2 n-i0( ∪j∈T bj ) = 2 i1( ∩j∈T bj ) , (19)
where (a) is due to [START_REF] Ma | New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes[END_REF]. To see this, note that

: b j, =0, j∈T K c = : b T ( )=0 K c ( 20
)
where b T = ∪j∈T b j .

B. Hamming Weight of the Sum of G N Rows

Let T ⊆ N be any subset of row indices of polar-like code generator matrix G N and g T be

g T = j∈T g j , (21)
and T w ⊆ T be any subset of T with |T w | = w. Then, the Hamming weight of g T is given by the following theorem:

Theorem 2. Let T ⊆ N be any subset of row indices of polarlike code generator matrix G N . Then, the Hamming weight of the sum of the rows g j , j ∈ T is given by

i 1 g T ) = |T | w=1 (-2) w-1 T w ∈T 2 i1 ¯ j∈T w bj . ( 22
)
Proof. Let us proceed by induction. For any two rows (g t , g m ), t, m ∈ N of G N , the Hamming weight of g j ⊕ g k is given by:

i 1 (g t ⊕ g m ) = i 1 (g t ) + i 1 (g m ) -2 • i 1 (g t ∩g m ). (23)
Note that (23) complies with the following expression

i 1 j∈T g j = j∈T i 1 (g j )+ |T | w=2 (-2) w-1 T w ∈T i 1 ∩j∈T w g j , (24)
where T = {t, m}. Now, assume that it holds for arbitrary T ⊂ N . By (23), i 1 (g T ) for T = T ∪ {t } can be written as

i 1 (g T ) = i 1 (g T ) + i 1 (g t ) -2 • i 1 (g T ∩g t ) = i 1 (g T ) + i 1 (g t ) -2 • i 1 ( j∈T (g j ∩g t )) (25)
Substituting ( 24) in ( 25) provides (26) on top of the next page where T w is any subset of T , i.e., T w ⊆ T , with |T w | = w. Substituting ( 16) in (24) results in (22) once noted that i 1 (g j ) = 2

i1( ¯ j∈T 1 b bj ) = 2 i1(bj ) for T 1 b = {b j }.

C. Code Design with Row Merging

The proposed code design is based on encoding some information bits with more than one row of the generator matrix G N such that some frozen bits are set to their corresponding preceding information bits, in a sense that it will be made clear in the following. For a given blocklength N = 2 n , we define the critical information lengths

k = k , ∈ [2, n -1], such that k = n p= n p . (27)
In order to guarantee the highest possible d min without dynamically frozen bits, the set of information bits, A, is chosen as the set of row indices which have the highest weights, which is called the Reed-Muller (RM) rule [START_REF] Arikan | Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels[END_REF] selection, that is to say, for k = k :

A = n p= N p ( 28
)
where

N p := {j ∈ N : i 1 (b j ) = p}.
Even though RM row selection guarantees a polar-like code with the highest d min with u F = 0, the number of codewords with Hamming weight d min is still high as it is illustrated in Table I. This table gives the number of codewords A d with the Hamming weight d, for the RM design, our proposed design (PD) and PAC codes. To decrease the number of low weight codewords, we propose a heuristic algorithm by which either the Hamming weight of the merged rows are increased or they remain unchanged but the positions of 1's in the merged rows are moved, which changes the regular structure of the positions of 1's of the information rows selected by the RM rule. For a given k , the Hamming weight of a row j ∈ N can be increased by merging it with a row m ∈ N -1 such that

i 1 (b j ∩b m ) = * , m > j (29)
where max{0, 2 -1 -n} ≤ * < -2. The merging with any row m such that

i 1 (b j ∩b m ) = -2, m ∈ N -1 , m > j (30)
results in a merged row with i 1 (g j ⊕ g m ) = 2 . However, by Theorems 1 and 2, the 1's at the indices P 1 (g j ∩g m ) are moved to the indices P 1 (g m ) \ P 1 (g j ∩g m ).

After having merged two rows (j, m) satisfying (29) one can still increase the Hamming weight by merging them with a third row t > j, t ∈ N -2 such that P 1 (b j ∩b m ∩b t ) = P 1 (b j ∩b m ) and

P 1 (b t )⊂(P 1 (b j ∩b m )∪P 0 (b j ∪b m )) if i 1 (b j ∪b m ) < n (31)
For (j, m) pairs satisfying (30), the third row t > j, t ∈ N -2 to increase the Hamming weight can be chosen as

P 1 (b t ∩b j )=P 1 (b t ∩b m ) and i 1 (b j ∩b m ∩b t )=i 1 (b j ∩b m ) -1 and P 1 (b t )⊂(P 1 (b j ∩b m )∪P 0 (b j ∪b m ))if i 1 (b j ∪b m )<n. ( 32
)
Based on these observations, for each * ∈ [max{0, 2 -1 -n}, -2] and for a given k = k , we implement Algorithm 1 to obtain pairs and triples to merge. The algorithm chooses the smallest m ∈ N -1 satisfying (29) or (30) starting from the smallest j ∈ N for any given * . Then, it chooses the smallest t ∈ N -2 satisfying (31) or (32) regarding the value of * . To obtain the codewords for any * ∈ [max{0, 2 -1 -n}, -2], the row merging operation is implemented at the encoding stage by choosing

u m (a) = h m (u 0 , • • • , u m-1 ) = u j , (33)
for m ∈ F, j ∈ A and j < m, where (a) is due to [START_REF] Zhang | CRC Code Design for List Decoding of Polar Codes[END_REF], or

h m (u 0 , • • • , u m-1 ) = h t (u 0 , • • • , u t-1 ) = u j , (34)
for (m, t) ∈ F, j ∈ A, j < m and j < t, if there exist corresponding merging pairs or triples, respectively. These operations only require reading from the memory in practice. Let C * denote the codebooks obtained by applying the row merging operations proposed by Algorithm 1. Then, the codebook with the best distance spectrum is chosen, i.e., the one with the lowest number of minimum weight codewords. If the distance spectrum of two codebooks are equal for the first w -1 smallest terms, w ≥ 2, then the one with the smallest number of codewords at the w-th term is chosen.

IV. NUMERICAL RESULTS

A. Distance Spectrum and FER Performance

Table I shows the number of codewords with different Hamming weights for blocklength N = 128 and k = 29, 64, and 99 by employing the same experiment as the one conducted in [START_REF] Li | An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check[END_REF]. For k = 99, k = 64 and k = 29 the minimum list size is 7×10 5 , 3×10 5 and 1.5×10 5 respectively. For PAC codes with k = 29 and k = 99, the generator polynomials have been optimized for the memory length m = 9, following the experimentin [START_REF] Li | An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check[END_REF] in order to produce the most favorable case of PAC codes. The chosen generator polynomial for k = 29 and k = 99 are [1 0 1 1 1 0 0 0 0 1] and [1 0 0 1 0 1 1 1 1 1], respectively. We observe that the proposed

i 1 (g T ) = j∈T i 1 (g j )+i 1 (g t ) -2 |T | w=2 (-2) w-2 T w ∈T i 1 ∩j∈T w g j + j∈T i 1 g j ∩g t + |T | w=2 (-2) w-1 T w ∈T i 1 ( ∩j∈T w g j ) ∩g t = j∈T i 1 (g j )+ |T | w=2 (-2) w-1 • T w ∈T i 1 ∩j∈T w g j + T w-1 ∈T i 1 ( ∩j∈T w-1 g j ) ∩g t + (-2) |T |-1 i 1 ( ∩j∈T g j ) = j∈T i 1 (g j )+ |T | w=2 (-2) w-1 • T ,w ∈T i 1 ∩j∈T ,w g j (26)
Algorithm 1: Row Merging Pairs and Triples for * .

Set M ← N -1 and W ← N -2 ; design allows to reduce the number of minimum weight codewords to the same order of magnitude of those obtained with optimized PAC codes. For (128, 64)-code, the numbers of codewords at Hamming weights 16 and 18 obtained with our proposal are even lower than for the ones obtained with PAC codes. For (128, 29)-code the chosen code does not have any codewords with Hamming weights 34. The number of codewords at certain Hamming weights are not computed due to the prohibitive amount of computations that they require.

for i from 1 to |N | do Set j ← N (i); Set M j ← {m : i 1 (b j ∩b m ) = * , m > j, m ∈ M}; if M j = ∅ then Set (j, m) ← (j, M j (1)) ; Set M ← M \ m; if * < -2 then Set W j,m ← {t : t > j, t ∈ W, satisf. (31)}; else Set W j,m ← {t : t > j, t ∈ W, satisf. (32)}; end Set (j, m, t) ← (j, m, W j,m (1)) ; Set W ← W \ t if W j,m = ∅; end end
Figures 1 and2 compare the FER achieved with our design with those obtained with RM and PAC codes for three different codes, i.e., (128, 29), (128, 64) and (128, 99), w.r.t. the uncoded energy per bit, i.e. E b /N 0 . Indeed, since codes with different rates are compared, E b /N 0 is a fairer base of comparison than SNR. This is the reason why one may observe a difference about 3 dB for R = 1/2 between our results compared to PAC codes in [START_REF] Arikan | From sequential decoding to channel polarization and back again[END_REF]. The performance of the codes are also compared w.r.t. the lower bound of the saddle-point approximation [START_REF] Anade | An Upper Bound on the Error Induced by Saddlepoint Approximations-Applications to Information Theory[END_REF] of the meta-converse bound [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF] in BI-AWGN channel. This bound, referred as converse saddle-point approximation (CSA) on Figures 1 and2, is a strict lower bound that cannot be outperformed. At the receiver side, SCL decoding with list size L = 256, that almost reaches the ML performance, is performed. It is seen that our method performs as well as the optimized Arikan's PAC codes for various information length k, with a lower complexity, as it will be explained in next Section. Moreover, FER achieved with our design is at most at 0.5dB from CSA at 10 -4 for k = 29, as it has been reported in Figure 2. As expected, PD outperforms RM codes due to the huge difference in the number of lowest weight codewords between the two schemes.

B. Complexity Comparison with PAC Codes

As stated in [START_REF] Yao | List Decoding of Arıkan's PAC Codes[END_REF], the difference of PAC encoding scheme from standard polar encoding is the rate-1 convolutional encoder that takes place before polar transformation, which is implemented through a length-m memory as follows [START_REF] Yao | List Decoding of Arıkan's PAC Codes[END_REF]:

u i = m j=0 c j v i-j , i ∈ N (35)
where c ∈ F 1×(m+1) 2

with c 0 = c m = 1 is the generator polynomial of the convolutional encoder, v A ∈ F 1×k 2 is the information vector and v F = 0 N -k-1 0 is the frozen vector, respectively. At the encoder side the PAC code requires O(m • N ) additional floating point operations besides extra memory requirement [START_REF] Arıkan | Systematic Encoding and Shortening of PAC Codes[END_REF]. Note that the extra computational complexity is equivalent to standard polar encoding for short block lengths if m ≈ log 2 N , which holds for the undertaken scenario, for instance.

The SCL decoding of PAC codes is implemented by adding a convolutional decoder after each SC decoder of list decoder as follows [START_REF] Yao | List Decoding of Arıkan's PAC Codes[END_REF]: On the contrary, the complexity of the proposed method is almost equivalent to the one of SCL decoder as it requires few memory readings due to the single argument of dynamic frozen functions as stated in (33) and (34).

vi = ûi - m j=1 c j vi-j , i ∈ A (36)

V. CONCLUSION

In this paper, we proposed a practical polar-like code construction method that performs similar, in terms of FER, to PAC codes, at short blocklengths but with a much smaller computational complexity. The method is based on the row merging that encodes some information bits with more than one rows of the polar encoding matrix. The proposed scheme has been constructed using the connection between the binary representation of row indices and the number of common 1bit positions of the corresponding rows in the polar matrix. Our design decreases the number of low weight codewords compared to the RM selection rule. The proposed scheme does not require more computational resources than classical SCL decoding and hence our scheme is a good candidate to implement polar codes in applications with stringent complexity constraints and with short packet transmission. As further work, the extension to higher blocklengths is under investigation since it may require increasing the list size decoding for proper performance.

1 EFigure 1 :

 11 Figure 1: FER obtained with our scheme (PD) w.r.t. PAC codes and CSA for N = 128, k = 64 and k = 99.

1 EFigure 2 :

 12 Figure 2: FER obtained with our scheme (PD) w.r.t. PAC codes and CSA for N = 128, k = 29.

Table I :

 I Number of low weight codewords in some codes.

		A 8	A 10	A 16 A 18 A 32 A 34
	(128, 29) RM	-	-	-	-10668 -
	(128, 29) PD	-	-	-	-	952	-
	(128, 29) PAC	-	-	-	-	348 360
	(128, 64) RM	-	-	94488 NC	NC	NC
	(128, 64) PD	-	-	2778 410	NC	NC
	(128, 64) PAC	-	-	3120 2696 NC	NC
	(128, 99) RM 188976	-	NC	NC	NC	NC
	(128, 99) PD	19226 > 10 5 NC	NC	NC	NC
	(128, 99) PAC 14432 > 10 5 NC	NC	NC	NC
	'-': no codeword at this Hamming weight; 'NC': not computed

In[START_REF] Miloslavskaya | Design of Short Polar Codes for SCL Decoding[END_REF], the name polar-like is used to represent original polar codes of[START_REF] Arikan | Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels[END_REF] with dynamic frozen bits.

In[START_REF] Trifonov | Polar codes with dynamic frozen symbols and their decoding by directed search[END_REF],[START_REF] Zhang | CRC Code Design for List Decoding of Polar Codes[END_REF] is defined as a function of the preceding information bits. Here, we give a general definition.


Kyangwi Malikidogo

Isidro Da Silva

Jean-François Morfin

Sara Lacerda

Laurent Barantin

Thierry Sauvage

Julien Sobilo

Stéphanie Lerondel

Éva Tóth

Célia S Bonnet
email: celia.bonnet@cnrs.fr

Patrick K Malikidogo

A cocktail of 165 Er( iii ) and Gd( iii ) complexes for quantitative detection of zinc using SPECT and MRI

Keywords: synthetic procedures, potentiometric titrations, relaxometric and photophysical measurements, production, purification, radiolabeling, MRI images, and uncertainty calculations

Molecular imaging seeks to characterize the physical-chemical state of tissues such as pH, the presence of enzymes, metabolites or ions. Magnetic Resonance Imaging (MRI) is particularly adapted to these applications as the MRI efficiency (relaxivity) of a contrast agent can be made specifically responsive to a given physiological parameter. In contrast, in nuclear imaging such as PET (positron emission tomography) or SPECT (single photon emission computed tomography), the signal generated by the imaging agent is independent of its environment, while its intensity is directly related to probe concentration. Combining high sensitivity nuclear images with the excellent spatial resolution of morphological MRI in dual imaging has become common practice. Bimodal imaging is also attractive to access to quantitative detection of physiological parameters.

1, 2 Indeed, local concentration of the MRI probe should be known in order to unambiguously attribute observed variations in MRI signal to changes in relaxivity and not in probe concentration. Although this potential has been recognized early on, reports on bimodal agents for quantitative detection are scarce and limited to pH mapping. F to circumvent the sensitivity problem. 3 The tedious synthesis required to introduce the nuclear reporter is a major drawback of this method. In the second approach, a cocktail can be tuned to respect the sensitivity of each imaging technique, but one has to make sure that the biodistribution of the two molecules is similar. Aime et al. took advantage of the similar chemical properties of Ln 3+ ions and used the mixture of a pHresponsive Gd 3+ complex and its radioactive Ln 3+ ( 166 Ho 3+ ) analogue. 4 With the production method used to obtain 166 Ho, This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins their sample was diluted in a high amount of 165 Ho. Although they managed to achieve pH detection, this approach would be unrealistic to detect biomarkers in M or even mM concentration, as the high quantity of bulk 165 Ho present would strongly limit the attainable MRI response. ErL analogue. 165 Er has an optimal half-life (t 1/2 = 10.36 h vs 26.6h for 166 Ho), 12 and emits Xradiation (50 KeV), suitable for SPECT imaging. 13 It is readily produced from 165 Ho (stable isotope with 100 % abundance) in a cyclotron and can therefore be separated from the parent isotope because it is a different element. Thus the cocktail can be prepared without dilution of the MRI reporter (Gd 3+ complex), contrary to the example above (see ESI).

4

The ligand L consists of a polyaminocarboxylate unit with a pyridine backbone (Py) to chelate the Ln 3+ , a linker, and a Zn 2+ binding unit (Fig. 1). The relatively good thermodynamic stability and kinetical inertness of LnPy complexes ensure their safe use in animal studies, as previously demonstrated. 14 The Ln 3+ has two inner sphere water molecules which are not replaced by physiological anions and confer high relaxivity to the Gd 3+ complex. 15 The Zn 2+ chelating moiety is a modified bis(pyridinylmethyl)amine (DPA) with an affinity for Zn 2+ (K d ca.

25 nM) 16 relevant for physiological applications. 5 We have introduced carboxylate functions on the DPA pyridines to potentially generate a change in the hydration number of the Gd 3+ and to minimize the formation of dimers upon Zn 2+ binding.

17

Ligand L was synthesized in 6 steps. After the formation of the lanthanide complexing unit, a lysine derivative spacer and the Zn 2+ complexing DPA unit were successively introduced. The full description is given in ESI.

Potentiometric titrations evidence the formation of a stable GdL complex, with log K GdL = 20.1(1), more than one order of magnitude higher than that of the parent Py. binding (Fig. 2a). Importantly for quantitative detection, the relaxivity increases linearly with increasing amounts of Zn 2+ (Fig. 2b). Both in the absence and in the presence of Zn 2+ , the relaxivity is independent of GdL concentration (figure S3) implying that a relaxivity vs. to previously reported Zn 2+ -responsive agents. In order to achieve quantitative detection of Zn 2+ , 165 Er 3+ was produced, purified and its radionuclide purity was assessed by -spectrometry with a HPGe detector (Canberra). It was then reacted with L to yield 165 ErL. The complex was obtained quantitatively after 10 min at room temperature (see ESI for detailed 165 Er production, purification and 165 ErL preparation).

A cocktail with a known GdL/ 165 ErL mole ratio (1/2.4x10 -7 ), respecting the relative sensitivity of MRI and SPECT detection, was prepared in HSA solution (0.6 mM, physiological concentration). This solution was then diluted to give five samples, to which unknown concentrations of Zn 2+ were added while keeping the HSA concentration constant at 0.6 mM for all samples. The activities were measured for all samples by spectrometry. Then the samples were imaged using a camera (Fig. 3b). The concentration of 165 Er 3+ was calculated from the signal intensity, giving access to the Gd 3+ concentration (by knowing the Gd/Er ratio). These Gd 3+ concentrations correspond within 6 % to those calculated according to the dilutions and the concentration of the Gd 3+ stock solution (which was determined by bulk magnetic susceptibility shifts (BMS)

22

). This is within the range of the experimental error (uncertainly) calculated to be ca. 16 % (see ESI). Finally, after total decay of the samples (ca. 1 to 2 weeks), a T 1 -weighted MR image was recorded (Fig. 3a) on a 1.5 T scanner (Signa HDxt, General Electric, Milwaukee, USA). The longitudinal relaxation times measured were translated into relaxivity using the Gd 3+ concentration calculated for each sample (see ESI). Based on a calibration curve recorded on the same MRI scanner and at the same temperature (Figure S6), the experimental Zn 2+ concentrations were calculated from the measured relaxivities (Table 1). The experimental error on the Zn 2+ concentrations was calculated to be 23% (see ESI). We also determined the Zn 2+ concentration of each sample by ICP, and a very good agreement was obtained (the difference between the concentration measured by ICP and determined via relaxivity measurements is maximum 13 %, within the experimental error). For comparison, between healthy and malignant prostate tissue for instance, Zn 2+ concentration varies from ca. 4 to 0.4 mM, representing a 90 % decrease in Zn 2+ concentration. 5 To take full benefit of this quantitative method, we are developing more efficient systems for Zn

Fig. 1 .

 1 Fig. 1. Structure and schematic representation of the Zn 2+ -responsive contrast agent. Here, we propose to use 165 Er 3+ and we demonstrate quantification of Zn 2+ with a cocktail of the Zn 2+ -responsive MRI agent GdL and its 165

  14 Mono-and bimetallic complexes are observed with Zn2+ , with affinity constants of log K ZnL = 16.5(1) and log K Zn2L = 9.9(1), respectively (see ESI for the full speciation). Log K GdL is nearly four orders of magnitude higher than log K ZnL demonstrating selectivity of the Py unit for Ln 3+ over Zn 2+ . The affinity of GdL for Zn 2+ can be estimated from log K Zn2L , and it is in the low nanomolar range, in accordance with Log K ZnL' = 9.08 of a model compound mimicking the Zn 2+ binding site (unpublished results). Upon addition of Zn 2+ to a GdL solution, no significant relaxivity changes were observed (see ESI). This excludes variation in the hydration number of GdL. It was corroborated with luminescence lifetime measurements on the corresponding Eu 3+ complexes, yielding two water molecules in the first coordination sphere of Gd 3+ (q = 2.0(3), see ESI for details) in the absence or presence of Zn 2+ . GdL contains three pyridine units which are known to interact with human serum albumin (HSA), and it was already demonstrated for DPA-containing agents that a relaxometric response to Zn 2+ can be obtained in the presence of HSA. 18, 19 Indeed, the agent alone binds only weakly to HSA, whereas in the presence of Zn(II), the binding is stronger resulting in higher relaxivity. Here, in the presence of HSA, GdL shows a relaxivity increase of ca. 25 % upon Zn 2+

Fig. 2 . 1 H

 21 Fig. 2. 1 H relaxivity measurements at pH = 7.4 (Hepes 0.1 M), at 20 MHz, in the presence of [GdL] = 0.30 mM: a) as a function of HSA, in the absence (), or in the presence of 1 eq. of Zn 2+ (), at 310 K; b) in the presence of [HSA] = 0.6 mM, with increasing amounts of Zn 2+ at 298 K.

Fig. 3 .

 3 Fig. 3. T1-weighted images of the five samples (T1 values are given in ms) at 1.5 T using a spin echo sequence with TE = 9 ms, TR = 100 ms; T1 were measured with variable TR from 20 to 2000 ms (a). Corresponding images from the -camera (activities are given in kBq) (b).

Table 1 .

 1 Zn 2+ concentration determined by the use of the bimodal cocktail of LnL (exp), and by ICP (th).  represents the difference between the two values. responsive imaging of other physiological cations, metabolites or any biomarker. Although the development of hybrid SPECT/MRI imagers is still in its infancy, a small number of systems are already under development, which could allow simultaneous detection by the two modalities.[23][24][START_REF] Cai | IEEE Nuclear Science Symposium conference record. Nuclear Science Symposium[END_REF] Finally, given its accessibility in

	[Gd	3+ ] is calculated based on the activities of	165 Er	3+ as described in the text.
			[Gd 3+ ] (mM)	[Zn 2+ ]exp	[Zn 2+ ]th	 (%)
						(mM)	(mM)
	Sample 1	0.30±0.05		0.33±0.08	0.32	3.0
	Sample 2	0.37±0.06		0.29±0.07	0.29	0
	Sample 3	0.44±0.07		0.54±0.13	0.49	9.3
	Sample 4	0.49±0.08		0.62±0.14	0.55	11.3
	Sample 5	0.53±0.09		0.10±0.03	0.094	6.0
	To sum up, we report a proof-of-concept of quantitative
	assessment of Zn	2+ concentration via combined MRI and SPECT
	detection using a Gd/	165 Er 3+ cocktail.	165 Er 3+ is easily produced
	and separated from adapted to medical imaging. The application of 165 Ho and its half-life (t ½ = 10.36h) is 165 Er 3+ could be
	generalized to quantitative cyclotron facilities, 165 Er 3+ is also an interesting surrogate to
	Gd

3+

to perform in vivo biodistribution studies of potential MRI agents by SPECT.

We thank Agnès Pallier for performing ICP measurements on the samples. CSB thanks the support of the French National Research Agency (grant ANR-13-JS07-0007).

Conflicts of interest

There are no conflicts to declare.

Notes and references