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The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]

We propose a successive cancellation list (SCL) encoding and decoding scheme for the Gelfand Pinsker (GP) problem based on the known nested polar coding scheme. It applies SCL encoding for the source coding part, and SCL decoding with a properly defined CRC for the channel coding part. The scheme shows improved performance compared to the existing method. A known issue with nested polar codes for binary dirty paper is the existence of frozen channel code bits that are not frozen in the source code. These bits need to be retransmitted in a second phase of the scheme, thus reducing the rate and increasing the required blocklength. We provide an improved bound on the size of this set, and on its scaling with respect to the blocklength, when the Bhattacharyya parameter of the test channel used for source coding is sufficiently large, or the Bhattacharyya parameter of the channel seen at the decoder is sufficiently small. The result is formulated for an arbitrary binary-input memoryless GP problem, since unlike the previous results, it does not require degradedness of the two channels mentioned above. Finally, we present simulation results for binary dirty paper and noisy write once memory codes. [ABSTRACT FROM AUTHOR]

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources. [ABSTRACT FROM AUTHOR]

In this paper, we establish some criteria to decide when a discrete memoryless channel admits a metric in such a way that the maximum likelihood decoding coincides with the nearest neighbor decoding. In particular, we prove a conjecture presented by M. Firer and J. L. Walker, establishing that every binary asymmetric channel admits a matched metric. [ABSTRACT FROM AUTHOR]

This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for $[m\times 2]$ -MRD codes with rank 2 this limiting proportion is close to $1/e$. [ABSTRACT FROM AUTHOR]

The zero-error capacity of a channel is defined as the maximum rate at which it is possible to transmit information with zero probability of error. In this paper, we settle all previously unsolved cases for the zero-error capacity of binary channels with one memory. [ABSTRACT FROM AUTHOR]

A classical method of constructing a linear code over \mathrm GF(q) with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over \mathrm GF(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented. [ABSTRACT FROM AUTHOR]

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes. [ABSTRACT FROM AUTHOR]

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(ƒ) = {Tr(aƒ(x) + bx)x∈Fqm*: a, b ∈ Fqm}, where q is a prime power, Fqm* = Fqm \ {0}, Tr is the trace function from Fqm to Fq, and ƒ(x) is a function from Fqm to Fqm with ƒ(0) = 0. Almost bent functions, quadratic functions and some monomials on F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we apply two-to-one functions over F2n in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x2t + x)e with gcd (t, n) = gcd(e, 2n−1) = 1. Based on the study of the Walsh transforms of those functions or their variants, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights, and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. Moreover, examples show that some codes in this paper have best-known parameters. [ABSTRACT FROM AUTHOR]

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^{n}$ ; let $r\geq 3$ be a positive integer satisfying $\gcd (r,n)=1$ and $\gcd \big (r,q(q^{2}-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

This paper generalizes the method proposed by Poulliat et al. for the determination of the optimal Galois field coefficients of a non-binary LDPC parity check constraint based on the binary image of the code. Optimal, or almost-optimal, parity check coefficients are given for check degree varying from 4 to 20 and Galois field varying from GF (64) up to GF (1024). For all given sets of coefficients, no codeword of Hamming weight two exists. A reduced complexity algorithm to compute the binary Hamming weight 3 of a parity check is proposed. When the number of sets of coefficients is too high for an exhaustive search and evaluation, a local greedy search is performed. Explicit tables of coefficients are given. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction. [ABSTRACT FROM AUTHOR]

Consider a channel with a given input alphabet size and input distribution. Our aim is to degrade or upgrade it to a channel with at most L output letters. Channel Q is degraded with respect to a channel W if Q can be obtained from W by processing the output of W. Upgrading is the inverse relation. This paper contains four main results. The first result, from which the paper title is derived, deals with the so called “greedy–merge” algorithm. We derive an upper bound on the reduction in mutual information between input and output, as a function of L. This upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy–merge is optimal in the power-law sense (i.e. the power of L). The other main results deal with upgrading. The second result shows that a certain sequence of channels, which was previously shown to be “hard” for degrading, displays the same hardness in the context of upgrading. That is, suppose we are given such a channel and a corresponding input distribution. If we upgrade (degrade) to a new channel with L output letters, we incur an increase (decrease) in mutual information between input and output. We show that a previously derived bound on the decrease in mutual information for the degrading case is also a lower bound on the increase for the upgrading case. The third result is an efficient algorithm for optimal upgrading, in the binary-input case. That is, we are given a channel and an input distribution. We must find an upgraded channel with L output letters, for which the increase in mutual information is minimal. We give a simple characterization of such a channel, which implies an efficient algorithm. The fourth result is an analog of the first result for the upgrading case when the input is binary. That is, we first present a sub-optimal algorithm for the setting considered in the third result. The main advantage of the sub-optimal algorithm is that it is amenable to analysis. We carry out the analysis and show that the increase incurred in mutual information is within a constant factor of the lower bound derived in the second result. [ABSTRACT FROM AUTHOR]

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes, which are binary codes after a Gray map from a subgroup of direct products of \mathbb Z2 , \mathbb Z4 , and Q8 , where Q8 is the quaternionic group. In a previous work, these codes were classified in five shapes. In this paper, we analyze the allowable range of values for the rank and dimension of the kernel, which depends on the particular shape of the code. We show that all these codes can be represented in a standard form, from a set of generators, which can help in understanding the characteristics of each shape. The main results we present are the characterization of Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes as a quotient of a semidirect product of \mathbb Z2 \mathbb Z4 -linear codes and the construction of Hadamard \mathbb Z2 \mathbb Z4 {Q}8 -codes with each allowable pair of values for the rank and dimension of the kernel. [ABSTRACT FROM AUTHOR]

We show an explicit (that is, efficient and deterministic) capacity approaching interactive coding scheme that simulates any interactive protocol under random errors with nearly optimal communication rate. Specifically, over the binary symmetric channel with crossover probability $\epsilon $ , our coding scheme achieves a communication rate of $1 - O(\sqrt {H({\epsilon })})$ , together with negligible $\exp (-\Omega (\epsilon ^{4}\,\,n/\log n))$ failure probability (over the randomness of the channel). A rate of $1 - \tilde \Theta (\sqrt {H({\epsilon })})$ is likely asymptotically optimal as a result of Kol and Raz (2013) suggests. Prior to this paper, such a communication rate was achievable only using randomized coding schemes [Kol and Raz (2013); Hauepler (2014)]. [ABSTRACT FROM AUTHOR]

In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the $q^\ell $ -incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding. By calculating the $q^\ell $ -incorrigible set distributions, the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions. For the ensemble of all $[n,k]$ linear codes, the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined. Furthermore, the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all $[n,nR]$ linear codes. [ABSTRACT FROM AUTHOR]

This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to Shannon et al. and which was shown to achieve the error exponent with a fixed number of codewords $\mathsf {M}$ , can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the $r$ -wise Hamming distance is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum $r$ -wise Hamming distance satisfies a generalized $r$ -wise Plotkin bound. The $r$ -wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the $r$ -wise Plotkin bound with equality for all $r$. In the second part of this paper, these insights are applied to a binary erasure channel with an arbitrary erasure probability 0 < $\delta$ < 1. An exact formula for the average error probability of an arbitrary (linear or nonlinear) code using maximum likelihood decoding is derived and shown to be expressible using only the $r$ -wise Hamming distance structure of the code. For a number of codewords $\mathsf {M}$ satisfying $\mathsf {M}\le 4$ and an arbitrary finite blocklength $n$ , the globally optimal codes (in the sense of minimizing the average error probability) are found. For $\mathsf {M}$ = 5 or $\mathsf {M}$ = 6 and an arbitrary finite blocklength $n$ , the optimal codes are conjectured. For larger $\mathsf {M}$ , observations regarding the optimal design are presented, e.g., that good codes have a large $r$ -wise Hamming distance structure for all $r$. Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed as accessible capacity, is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryless channel, the accessible capacity can be characterized as the difference between the channel capacity and the mutual information rate of the coded payload link, which can be numerically evaluated for very short payload codes. For a general payload code, we present a simple lower bound on the accessible capacity, given by the channel capacity minus the coding rate of the payload code. We then focus on the scenarios where low-density parity-check (LDPC) codes are implemented for the payload link. We propose to transmit extra bits by random superposition for encoding, and exhaustive search (with the aid of statistical learning) for decoding. We further propose, by establishing an auxiliary channel (called syndrome channel) induced from “zero-forcing” over the binary field, to transmit extra bits with structured codes such as repetition codes and first-order Reed-Muller (RM) codes. Numerical results show that up to 60 extra bits can be reliably transmitted along with a rate-1/2 LDPC code of length 8064. [ABSTRACT FROM AUTHOR]

Deletion errors are notoriously difficult to correct. The capacity of the binary deletion channel is not yet fully known, and there is no completely satisfactory solution even to the specific problem of correcting only two deletion errors. In this paper, we show that polarization theory, along with the powerful methods of polar coding and successive-cancellation decoding, can be extended to channels with deletion errors. Our results further generalize to channels corrupted by insertion errors and/or combinations of insertions and deletions. It was recently proposed to correct deletion errors using polar codes designed for the BEC, precoded with a sufficiently powerful CRC code. In this approach, in order to correct ${d}$ deletions in a codeword of length ${n}$ , successive-cancellation decoding is run separately for each of the $\binom {n}{d}$ possible deletion patterns, while treating the deleted symbols as erasures. The resulting complexity of decoding is ${O}\bigl ({n}^{d+1}\log {n}\bigr)$. In contrast, we develop herein a natural generalization of successive-cancellation decoding to channels with deletion errors. Our algorithm is based on the recursive structure of polar codes, processing the channel output in ${m} = \log _{2} {n}$ decoding layers, just like the conventional successive-cancellation decoder. Every node in each decoding layer propagates its uncertainty about the deletion pattern to the nodes in the next layer and, eventually, the correct deletion pattern becomes visible at the decoder output, with high probability. This makes it possible to consider at most $\binom {d+2}{2}$ scenarios at each node, and the overall decoding complexity is only ${O}\bigl ({d}^{3} {n} \log {n}\bigr)$ , which scales polynomially rather than exponentially with the number of deletions. Based on the proposed algorithm, we also investigate the channel polarization phenomenon. We are able to prove weak polarization where the number of deletions ${d}$ is small, i.e. ${d}={o}({n})$. We also conjecture the strong polarization for when ${d}={O}(1)$ and provide some evidence to support this conjecture. The strong polarization, if proven, implies that our coding scheme achieves the capacity of any binary-input symmetric channel which furthermore introduces a constant number of deletions. [ABSTRACT FROM AUTHOR]

This paper studies a large random matrix system (LRMS) model involving an arbitrary signal distribution and forward error control (FEC) coding. We establish an area property based on the approximate message passing (AMP) algorithm. Under the assumption that the state evolution for AMP is correct for the coded system, the achievable rate of AMP is analyzed. We prove that AMP achieves the constrained capacity of the LRMS with an arbitrary signal distribution provided that a matching condition is satisfied. As a byproduct, we provide an alternative derivation for the constraint capacity of an LRMS using a proved property of AMP. We discuss realization techniques for the matching principle of binary signaling using irregular low-density parity-check (LDPC) codes and provide related numerical results. We show that the optimized codes demonstrate significantly better performance over un-matched ones under AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate (BER) performance within 1 dB from the constrained capacity limit is observed. [ABSTRACT FROM AUTHOR]

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function’s resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums at each of the non-zero time-shifts within a certain region, called the zero correlation zone (ZCZ). ZCPs are categorised into two types: Type-I ZCPs and Type-II ZCPs. Type-I ZCPs have the ZCZ around the in-phase position and Type-II ZCPs have the ZCZ around the end-shift position. Till now only a few constructions of Type-II ZCPs are reported in the literature, and all have lengths of the form 2m ± 1 or N + 1 where N = 2a 10b 26c and a, b, c are non-negative integers. In this paper, we propose a recursive construction of ZCPs based on concatenation of sequences. Inspired by Turyn’s construction of Golay complementary pairs, we also propose a construction of Type-II ZCPs from known ones. The proposed constructions can generate optimal Type-II ZCPs with new flexible parameters and Z-optimal Type-II ZCPs with any odd length. In addition, we give upper bounds for the PMEPR of the proposed ZCPs. It turns out that our constructions lead to ZCPs with low PMEPR. [ABSTRACT FROM AUTHOR]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context. [ABSTRACT FROM AUTHOR]

In this paper, two different Gray-like maps from $Z_{p}^\alpha \times Z_{p^{k}}^\beta $ , where $p$ is prime, to $Z_{p}^{n}$ , $n={\alpha +\beta p^{k-1}}$ , denoted by $\varphi $ and $\Phi $ , respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_{p} Z_{p^{k}}$ -additive code, and $C^\bot $ is its dual, then the weight enumerators of the image $p$ -ary codes $\varphi (C)$ and $\Phi (C^\bot)$ are formally dual. This is a partial generalization of [D. S. Krotov, On $Z_{2^{k}}$ -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532–1537], and the result is generalized to odd characteristic $p$ and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed $Z_{p} Z_{p^{2}}\ldots Z_{p^{k}}$ alphabet is given. [ABSTRACT FROM AUTHOR]

Lattice codes are elegant and powerful structures that not only achieve the capacity of the additive white Gaussian noise (AWGN) channel but are also a key ingredient to many multiterminal schemes that exploit linearity properties. However, constructing lattice codes that can realize these benefits with low complexity is still a challenging problem. In this paper, efficient encoding and decoding algorithms are proposed for multilevel binary low-density parity-check (LDPC) lattices constructed via Construction $\text{D}'$ whose complexity is linear in the total number of coded bits. Moreover, a generalization of Construction $\text{D}'$ is proposed, which relaxes some of the nesting constraints on the parity-check matrices of the component codes, leading to a simpler and improved design. Based on this construction, low-complexity multilevel LDPC lattices are designed whose performance under multistage decoding is comparable to that of polar lattices and close to that of low-density lattice codes on the power-unconstrained AWGN channel. [ABSTRACT FROM AUTHOR]

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $\sigma $ complementary dual ($\sigma $ -LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, $\sigma $ -LCD codes can also be used to construct LCP of codes. We show that for $q > 2$ , all $q$ -ary linear codes are $\sigma $ -LCD, and for every binary linear code $\mathcal C$ , the code $\{0\}\times \mathcal C$ is $\sigma $ -LCD. Furthermore, we study deeply $\sigma $ -LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of $\sigma $ -LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good $\sigma $ -LCD GQC codes. Finally, we focus on $\sigma $ -LCD abelian codes and prove that all abelian codes in a semi-simple group algebra are $\sigma $ -LCD. The results derived in this paper extend those on the classical LCD codes and show that $\sigma $ -LCD codes allow the construction of LCP of codes more easily and with more flexibility. [ABSTRACT FROM AUTHOR]

In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with $w_{\min }/w_{\max } \leq 1/2$ from a generic construction, where $w_{\min }$ and $w_{\max }$ , respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined. [ABSTRACT FROM AUTHOR]

We consider the problem of constructing binary codes for correcting deletions that are localized within certain parts of the codeword that are unknown a priori. The model that we study is when $\delta \leq w$ deletions are localized in a window of size $w$ bits. These $\delta $ deletions do not necessarily occur in consecutive positions, but are restricted to the window of size $w$. The localized deletions model is a generalization of the bursty model, in which all the deleted bits are consecutive. In this paper, we construct new explicit codes for the localized model, based on the family of Guess & Check codes which was previously introduced by the authors. The codes that we construct can correct, with high probability, $\delta \leq w$ deletions that are localized in a single window of size $w$ , where $w$ grows with the block length. Moreover, these codes are systematic; have low redundancy; and have efficient deterministic encoding and decoding algorithms. We also generalize these codes to deletions that are localized within multiple windows in the codeword. [ABSTRACT FROM AUTHOR]

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover’s enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary $n$ in a straightforward way. [ABSTRACT FROM AUTHOR]

$\mathbb {Z}_{2^{s}}$ -additive codes are subgroups of $\mathbb {Z}^{n}_{2^{s}}$ , and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{2^{s}}$ -linear code is a binary code (not necessarily linear) which is the Gray map image of a $\mathbb {Z}_{2^{s}}$ -additive code. We consider $\mathbb {Z}_{2^{s}}$ -additive simplex codes of type $\alpha $ and $\beta $ , which are a generalization over $\mathbb {Z}_{2^{s}}$ of the binary simplex codes. These codes are related to the $\mathbb {Z}_{2^{s}}$ -additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding $\mathbb {Z}_{2^{s}}$ -linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, $\mathbb {Z}_{2^{s}}$ -additive MacDonald codes are defined for $s>2$ , and equivalent results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-ADMM iteration scales linearly with LDPC code’s length and the size of the considered Galois field. Moreover, the decoder based on the Constant-Weight embedding technique satisfies the favorable property of codeword symmetry. Simulation results demonstrate their effectiveness in comparison with state-of-the-art LDPC decoders. [ABSTRACT FROM AUTHOR]

The well-known Huffman algorithm is an elegant approach to solve the basic problem of finding the optimal code among those with Kraft sums smaller than or equal to 1. In this paper, an extended problem is investigated, where the Kraft sum of the usable codes is limited to a given real number $\gamma $ , $0<\gamma <1$ . Unlike the case $\gamma =1$ (i.e., Huffman coding), for $\gamma <1$ , the Kraft sum of the optimal code is not necessarily equal to $\gamma $ . However, we show that the Kraft sum of the optimal code lies in a finite set of rational numbers which depend on the value of $\gamma $ . This motivates us to first devise a recursive algorithm to obtain the optimal codelengths with Kraft sum equal to $\gamma '$ , where $\gamma '$ has a finite binary representation and $0<\gamma '<1$ . To do so, the set of $N$ -tuple codelength vectors with a specific Kraft sum $\gamma '$ is characterized. The binary representation of $\gamma $ plays a fundamental role in all problems that arise for $\gamma <1$ . [ABSTRACT FROM PUBLISHER]

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds. [ABSTRACT FROM PUBLISHER]

This paper studies the parameters for which binary Reed–Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and, in particular, when can they achieve capacity for these two classical channels. Necessarily, this paper also studies the properties of evaluations of multivariate $GF(2)$ polynomials on the random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about the square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$ , the matrix whose rows are the truth tables of all the monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$ , which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate, we construct a new code $C'$ obtained by tensorizing $C$ , such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$ , then $C'$ can recover from errors in $S$ . Specializing this to the RM codes and using our results for erasures imply our result on the unique decoding of the RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent bounds from constant degree to linear degree polynomials. [ABSTRACT FROM PUBLISHER]

A primitive $k$ -batch code encodes a string $\boldsymbol {x}$ of length $n$ into a string $\boldsymbol {y}$ of length $N$ , such that each multiset of $k$ symbols from $\boldsymbol {x}$ has $k$ mutually disjoint recovering sets from $\boldsymbol {y}$. In this paper, we discuss new constructions of binary primitive batch codes. First, we develop novel explicit and random coding constructions of linear primitive batch codes based on finite geometries. Second, a new explicit coding construction of binary primitive batch codes based on bivariate lifted multiplicity codes is provided. For any $k=n^ \varepsilon $ with $\varepsilon \in (0,0.47)\setminus \{1/5,1/4\}$ , our proposed codes have a better trade-off between the redundancy and the parameters $k,n$ than previously known batch codes. [ABSTRACT FROM AUTHOR]

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb {F}_{q}$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536–6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb {F}_{q}$ , we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536–6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067–7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes. [ABSTRACT FROM AUTHOR]

In this paper, a general binary-input binary-output channel is investigated in the presence of feedback and input constraints. The feedback capacity and the optimal input distribution of this setting are calculated for the case of an $(1,\infty )$ -RLL input constraint, that is, the input sequence contains no consecutive ones. These results are obtained via explicit solution of an equivalent dynamic programming optimization problem. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The posterior matching scheme for our input-constrained setting is shown to achieve capacity using two new ideas: history bits, which captures the memory embedded in our setting, and message-interval splitting, which eases the analysis of the scheme. Additionally, in the special case of an S-channel, we give a very simple zero-error coding scheme that is shown to achieve capacity. For the input-constrained binary symmetric channel, we show using our capacity formula that feedback increases capacity when the cross-over probability is small. [ABSTRACT FROM AUTHOR]

Non-overlapping codes are a set of codewords in $\bigcup _{n \ge 2} \mathbb {Z}_{q}^{n}$ , where $\mathbb {Z}_{q} = \{0,1, {\dots },q-1\}$ , such that the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for variable-length codes, a codeword does not contain any other codeword as a subword. In this paper, we investigate a generic method to generalize binary codes to $q$ -ary ones for $q > 2$ , and analyze this generalization on the two constructions given by Levenshtein (also by Gilbert; Chee, Kiah, Purkayastha, and Wang) and Bilotta, respectively. The generalization on the former construction gives large non-expandable fixed-length non-overlapping codes whose size can be explicitly determined; the generalization on the latter construction is the first attempt to generate $q$ -ary variable-length non-overlapping codes. More importantly, this generic method allows us to utilize the generating function approach to analyze the cardinality of the underlying $q$ -ary non-overlapping codes. The generating function approach not only enables us to derive new results, e.g., recurrence relations on their cardinalities, new combinatorial interpretations for the constructions, and the superior limit of their cardinalities for some special cases, but also greatly simplifies the arguments for these results. Furthermore, we give an exact formula for the number of fixed-length words that do not contain the codewords in a variable-length non-overlapping code as subwords. This thereby solves an open problem by Bilotta and induces a recursive upper bound on the maximum size of variable-length non-overlapping codes. [ABSTRACT FROM AUTHOR]

Linear codes over $\mathbb {Z}_{p^{s}}$ of length $n$ are subgroups of $\mathbb {Z}_{p^{s}}^{n}$. These codes are also called $\mathbb {Z}_{p^{s}}$ -additive codes and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{p^{s}}$ -linear code is a code over $\mathbb {Z}_{p}$ , not necessarily linear, which is the generalized Gray map image of a $\mathbb {Z}_{p^{s}}$ -additive code. In 2015, a systematic encoding was found for $\mathbb {Z}_{4}$ -linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any $\mathbb {Z}_{p^{s}}$ -linear code with $s\geq 2$ and $p$ prime. We also describe a permutation decoding method for any systematic code over $\mathbb {Z}_{p}$ , not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by ${t}$ -criss-cross deletions defined as any combination of ${t_{\mathrm {r}}}$ row and ${t_{\mathrm {c}}}$ column deletions such that ${t_{\mathrm {r}}}+ {t_{\mathrm {c}}}= {t}$ for a given $t$. We show an equivalence between correcting ${t}$ -criss-cross deletions and ${t}$ -criss-cross insertions and show that a code correcting ${t}$ -criss-cross insertions/deletions has redundancy at least ${t} n + {t}\log n - \log ({t}!)$. Then, we present an existential construction of a ${t}$ -criss-cross insertion/deletion correcting code with redundancy bounded from above by ${t} n + \mathcal {O}({t}^{2} \log ^{2} n)$. The main ingredients of the presented code construction are systematic binary ${t}$ -deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient. [ABSTRACT FROM AUTHOR]

This paper presents an extension of the Elias bound on the minimum distance of codes for discrete alphabets with general, possibly infinite valued, distances. The bound is obtained by combining a previous extension of the Elias bound, introduced by Blahut, with an extension of a bound previously introduced by the author which builds upon ideas of Gallager, Lovász, and Marton. The result can in fact be interpreted as a unification of the Elias bound and of Lovász’s bound on graph (or zero-error) capacity, both being recovered as particular cases of the one presented here. Previous extensions of the Elias bound by Berlekamp, Blahut, and Piret are shown to be included as particular cases of our bound. Applications to the reliability function are then discussed. [ABSTRACT FROM AUTHOR]

In this paper, code decompositions (a.k.a. code nestings) are used to design binary polarization kernels. The proposed kernels are in general nonlinear. They provide a better polarization exponent than the previously known kernels of the same dimensions. In particular, nonlinear kernels of dimensions 14, 15, and 16 are constructed and are shown to have optimal asymptotic error-correction performance. The optimality is proved by showing that the exponents of these kernels achieve a new upper bound that is developed in this paper. [ABSTRACT FROM AUTHOR]

Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

Let $m$ be an arbitrary positive integer and $D_{8m}$ be the dihedral group of order $8m$ , i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^{2}=1, yxy=x^{-1}\rangle $. Left ideals of the dihedral group algebra $\mathbb {F}_{2}[D_{8m}]$ are called binary left dihedral codes of length $8m$ , and abbreviated as binary left $D_{8m}$ -codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$ -codes. These codes make up an important class of self-dual binary $[8m,4m]$ -codes such that the dihedral group $D_{8m}$ is necessarily a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left $D_{8m}$ -codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary $[{48,24,12}]$ -code and an extremal self-dual binary $[{56,28,12}]$ -code from self-dual binary left $D_{48}$ -codes and left $D_{56}$ -codes respectively. [ABSTRACT FROM AUTHOR]

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting $[[21{s}+5,3,16{s}+3;21{s}+2]]$ code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code. [ABSTRACT FROM AUTHOR]