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As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length n over Fq is always relative to an n-th primitive root of unity β in an extension field of Fq, and is called a dually-BCH code if its dual is also a BCH code relative to the same β. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances δ will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances δ. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

The Binary Iterative Hard Thresholding (BIHT) algorithm is a popular reconstruction method for one-bit compressed sensing due to its simplicity and fast empirical convergence. There have been several works about BIHT but a theoretical understanding of the corresponding approximation error and convergence rate still remains open. This paper shows that the normalized version of BIHT (NBHIT) achieves an approximation error rate optimal up to logarithmic factors. More precisely, using $m$ one-bit measurements of an $s$-sparse vector $x$, we prove that the approximation error of NBIHT is of order $O \left(1 \over m \right)$ up to logarithmic factors, which matches the information-theoretic lower bound $\Omega \left(1 \over m \right)$ proved by Jacques, Laska, Boufounos, and Baraniuk in 2013. To our knowledge, this is the first theoretical analysis of a BIHT-type algorithm that explains the optimal rate of error decay empirically observed in the literature. This also makes NBIHT the first provable computationally-efficient one-bit compressed sensing algorithm that breaks the inverse square root error decay rate $O \left(1 \over m^{1/2} \right)$., Comment: Submitted to a journal

We consider a complex Gaussian wiretap channel with finite-resolution analog-to-digital converters (ADCs) at both the legitimate receiver and the eavesdropper. For this channel, we show that a positive secrecy rate is always achievable as long as the channel gains at the legitimate receiver and at the eavesdropper are different, regardless of the quantization levels of the ADCs. For the achievability, we first consider the case of one-bit ADCs at the legitimate receiver and apply a binary input distribution where the two input points have the same phase when the channel gain at the legitimate receiver is less than that at the eavesdropper, and otherwise the opposite phase. Then the result is generalized for the case of arbitrary finite-resolution ADCs at the legitimate receiver by translating the input distribution appropriately. For the special case of the real Gaussian wiretap channel with one-bit ADCs at both the legitimate receiver and the eavesdropper, we show that our choice of input distribution satisfies a necessary condition of optimal distributions for Wyner codes., This paper was submitted to IEEE Transactions on Information Theory, and was presented in part at ITW 2019 in Visby, Sweden in Aug 2019

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some t-intersecting families related to the well-known Erd\"{o}s-Ko-Rado theorem, are constructed.

We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$ . Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes.

Let $1 be the ordered primitive roots modulo ${p}$ . We study the pseudorandomness of the binary sequence $(\text {s}_ {n})$ defined by ${s}_{n}\equiv \text {g}_{n+1}+{g}_{n+2}\bmod 2,\,\, {n}=0,1,\ldots $ In particular, we study the balance, linear complexity and 2-adic complexity of $({s}_{n})$ . We show that for a typical ${p}$ the sequence $({s}_{n})$ is quite unbalanced. However, there are still infinitely many p such that $({s}_{n})$ is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and 2-adic complexity of $({s}_{n})$ and state sufficient conditions for attaining their maximums. Hence, for carefully chosen ${p}$ , these sequences are attractive candidates for cryptographic applications.

Bazzi and Mitter [3] showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with good mathematical properties are asymptotically good. If the characteristic of the field is even, we construct asymptotically good self-dual dihedral group codes. If the characteristic of the filed is odd, we construct both the asymptotically good self-orthogonal dihedral group codes, and the asymptotically good LCD dihedral group codes.

Compressed sensing is an emerging theory of signal processing and it has wide applications in many frontier fields. The construction of the measurement matrices is still a central problem in compressed sensing. In this paper, two types of deterministic constructions of binary measurement matrices are presented via unitary geometry. Then, the lower bounds of the spark of unitary geometry measurement matrices are theoretically analyzed, and an asymptotic comparison between unitary geometry measurement matrices and projective geometry measurement matrices is given via the worst-case recovery capability. After that, a clipping-embedding operation is proposed for binary matrices to generate measurement matrices with more sizes, which can strongly extend the applicability of the deterministic binary matrices in practice. Finally, simulation results demonstrate that the performance of our measurement matrices is comparable to, sometimes even better than, that of the corresponding Gaussian random matrices under OMP and BP.

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 so-called Turbo approximate message passing (Turbo-AMP) algorithm. Under the assumption that the state evolution for AMP is correct for the coded system, the achievable rate of Turbo-AMP is analyzed. We prove that Turbo-AMP achieves the constraint 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 optimized codes demonstrate significantly better performance over un-matched ones under Turbo-AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate (BER) performance within 1 dB from the constrained capacity limit is observed., Accepted by IEEE Trans. on Information Theory, 16 pages, 17 figures. [A low-complexity capacity optimal AMP is designed for large random linear systems with arbitrary input distributions based on matched FEC coding.]

We consider a query-based data acquisition problem for binary classification of unknown labels, which has diverse applications in communications, crowdsourcing, recommender systems and active learning. To ensure reliable recovery of unknown labels with as few number of queries as possible, we consider an effective query type that asks "group attribute" of a chosen subset of objects. In particular, we consider the problem of classifying $m$ binary labels with XOR queries that ask whether the number of objects having a given attribute in the chosen subset of size $d$ is even or odd. The subset size $d$, which we call query degree, can be varying over queries. We consider a general noise model where the accuracy of answers on queries changes depending both on the worker (the data provider) and query degree $d$. For this general model, we characterize the information-theoretic limit on the optimal number of queries to reliably recover $m$ labels in terms of a given combination of degree-$d$ queries and noise parameters. Further, we propose an efficient inference algorithm that achieves this limit even when the noise parameters are unknown., Accepted to IEEE Transactions on Information Theory. 37 pages, 9 figures

An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this article, we investigate codes that correct either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with $\lceil {\log \text {n}}\rceil+\text {O}(\log \log \text {n})$ redundancy bits, while the other corrects a single indel with $\lceil {\log \text {n}}\rceil+2$ redundant bits. These two encoders are order-optimal . The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the $\mathtt {GC}$ -balanced constraint and require that exactly half of the symbols of any DNA codeword to be either $\mathtt {C}$ or $\mathtt {G}$ . In particular, via a modification of Knuth’s balancing technique, we provide a linear-time map that translates binary messages into $\mathtt {GC}$ -balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of $\mathtt {GC}$ -balanced codes that correct a single indel or a single edit.

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $\mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n \not\equiv 6,13,14,21,22,28,29 \pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n \equiv 0,3,4,5,10,11,12 \pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code., 19 pages; to appear in IEEE Trans. Inform. Theory

One-dimensional (1-D) Golay complementary pair (GCP) and its extension Z-complementary pair (ZCP) with additional zero autocorrelation zone property have been widely studied in the literature. Recently, the investigation of 1-D GCPs have been extended to two-dimensional (2-D) and higher dimensional Golay complementary array pairs (GCAPs). In this article, the concept of zero correlation zone (ZCZ) is applied to 2-D GCAP and the 2-D Z-complementary array pair (ZCAP) is presented. Several constructions of ZCAPs are proposed in this article and the existence of ZCAPs is discussed as well. Based on the proposed constructions, 2-D ZCAPs are shown to exist for all sizes. Furthermore, an upper bound on the rectangular ZCZ size of the 2-D ZCAP is derived when both dimensions are odd. The proposed constructions can include ZCAPs which achieve the upper bound.

We consider the class of channels formed from the concatenation of a deletion channel and a finite-state channel. For this class of channels, we show that the operationally-defined capacity is equal to the stationary capacity. We also show that the stationary capacity can be approached by a sequence of Markov processes with increasing Markovian orders. As a by-product, we show that the polar coding scheme proposed by Tal, Pfister, Fazeli, and Vardy [arxiv: 1904.13385 (2019)] achieves the capacity of the binary deletion channel.

In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of $\mathbb {F}_{2}$ . Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.

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.

The objective of a genome-wide association study (GWAS) is to associate subsequences of individuals’ genomes to the observable characteristics called phenotypes (e.g., high blood pressure). Motivated by the GWAS problem, in this paper we introduce the information-theoretic problem of associated subsequence retrieval , where a dataset of N (possibly high-dimensional) sequences of length G, and their corresponding observable (binary) characteristics is given. The sequences are chosen independently and uniformly at random from $\mathcal {X}^{\text {G}}$ , where $\mathcal {X}$ is a finite alphabet. The observable (binary) characteristic is only related to a specific unknown subsequence of length $L$ of the sequences, called associated subsequence . For each sequence, if the associated subsequence of it belongs to a universal finite set, then it is more likely to display the observable characteristic (i.e., it is more likely that the observable characteristic is one). The goal is to retrieve the associated subsequence using a dataset of N sequences and their observable characteristics. We demonstrate that as the parameters N, G, and L grow, a threshold effect appears in the curve of probability of error versus the rate which is defined as ${{\it\text { Gh}}(\text {L}/\text {G})}/{\text {N}}$ , where $\text {h}(\cdot )$ is the binary entropy function. This effect allows us to define the capacity of associated subsequence retrieval. We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the $\epsilon $ -error-rate.

In this article, we consider a secure one-helper distributed source coding system, with action-dependent side information. Specifically, the main encoder observes an independent and identically distributed (i.i.d.) source $X^{\textit{n}}$ and wishes to compress this source to the decoder. The helper compresses the correlated source $Y^{\textit{n}}$ to the decoder. The eavesdropper, having access to the information bits sent by the main encoder, can observe side information $Z^{\textit{n}}$ as well. Besides, side information $Z^{\textit{n}}$ and correlated source $Y^{\textit{n}}$ can be influenced by a cost-constrained action sequence taken by the decoder. This class of problems can be seen as an nontrivial extension of the secure one-helper source coding with additional cost-constrained actions. The purpose of this article is to study the impact of the helper on achievable rate-distortion-cost-leakage. An inner bound of the achievable rate-distortion-cost-leakage is provided for the discrete memoryless setting. This inner bound is further shown to be tight when $Y^{\textit{n}}$ is required to be reconstructed losslessly, and equivalent to the inner bound of Villard et al . when there is no action taken at the decoder. Numerical examples on binary symmetric sources are also discussed in this article.

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.

We study a guessing game where Alice holds a discrete random variable $X$ , and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole’s answer is however unreliable, and is incorrect with probability $\epsilon $ . We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities – this question simultaneously minimizes all the guessing moments for any value of $\epsilon $ . In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$ -ary question, and then observe Carole’s answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob’s question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$ ) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$ -ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$ .

It was demonstrated that, as a nonlinear implementation of Slepian-Wolf Coding, Distributed Arithmetic Coding (DAC) outperforms traditional Low-Density Parity-Check (LPDC) codes for short code length and biased sources. This fact triggers research efforts into theoretical analysis of DAC. In our previous work, we proposed two analytical tools, Codebook Cardinality Spectrum (CCS) and Hamming Distance Spectrum, to analyze DAC for independent and identically-distributed (i.i.d.) binary sources with uniform distribution. This article extends our work on CCS from uniform i.i.d. binary sources to biased i.i.d. binary sources. We begin with the final CCS and then deduce each level of CCS backwards by recursion. The main finding of this article is that the final CCS of biased i.i.d. binary sources is not uniformly distributed over [0, 1). This article derives the final CCS of biased i.i.d. binary sources and proposes a numerical algorithm for calculating CCS effectively in practice. All theoretical analyses are well verified by experimental results.

A tiling of the $n$-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the $n$-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the concatenation of two smaller ones., Comment: arXiv admin note: text overlap with arXiv:1903.05951

In this paper we study array-based codes over graphs for correcting multiple node failures. These codes have applications to neural networks, associative memories, and distributed storage systems. We assume that the information is stored on the edges of a complete undirected graph and a node failure is the event where all the edges in the neighborhood of a given node have been erased. A code over graphs is called $\rho $ -node-erasure-correcting if it allows to reconstruct the erased edges upon the failure of any $\rho $ nodes or less. We present a binary optimal construction for double-node-erasure correction together with an efficient decoding algorithm, when the number of nodes is a prime number. Furthermore, we extend this construction for triple-node-erasure-correcting codes when the number of nodes is a prime number and two is a primitive element in $\mathbb {Z}_{n}$ . These codes are at most a single bit away from optimality.

In this paper, we study a class of binary nonlinear feedback shift register sequences generated by cascaded feedback registers, one is an LFSR and the other one generates a de Bruijn sequence. The cycle structure (in particular, the initial state of each cycle) is determined by solving a system of linear equations. As an application, we can generate de Bruijn sequences of large period algorithmically.

Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\Bbb F_2+u\Bbb F_2$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes., Comment: 17 pages, To appear in IEEE IT

It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths $15$, $16$, $30$, $31$, $32$, $45$--$51$, and $\ge 60$., Comment: 12 pages, 4 tables

Write-once memory (WOM) is a storage device consisting of binary cells that can only increase their levels. A $t$ -write WOM code is a coding scheme that makes it possible to write $t$ times to a WOM without decreasing the levels of any of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory during the $t$ writes and the number of cells. It is known that the maximum possible sum-rate of a $t$ -write WOM code is $\log (t+1)$ . This is also an achievable upper bound, both by information-theoretic arguments and through explicit constructions. While existing constructions of WOM codes are targeted at the sum-rate, we consider here two more figures of merit. The first one is the complexity of the encoding and decoding maps. The second figure of merit is the convergence rate , defined as the minimum code length $n(\delta)$ required to reach a point that is $\delta $ -close to the capacity region. One of our main results in this paper is a capacity-achieving construction of two-write WOM codes which has polynomial encoding/decoding complexity while the block length $n(\delta)$ required to be $\delta $ -close to capacity is significantly smaller than existing constructions. Using these two-write WOM codes, we then obtain three-write WOM codes that approach a sum-rate of 1.809 at relatively short block lengths. We also provide several explicit constructions of finite length three-write WOM codes; in particular, we achieve a sum-rate of 1.716 by using only 93 cells. Finally, we modify our two-write WOM codes to construct $\epsilon $ -error WOM codes of high rates and small probability of failure.

In this correspondence, we correct an erroneous result on the achievability part of the Renyi common information with order $1+s\in (1,2]$ in (L. Yu and V. Y. F. Tan, “Wyner’s common information under Renyi divergence measures,” IEEE Trans. Inf. Theory, vol. 64, no. 5, pp. 3616–3632, May 2018). The new achievability result (upper bound) of the Renyi common information no longer coincides with Wyner’s common information. We also provide a new converse result (lower bound) in this correspondence for the Renyi common information with order $1+s\in (1,\infty]$ . Numerical results show that for doubly symmetric binary sources, the new upper and lower bounds coincide for the order $1+s\in (1,2]$ and they are both strictly larger than Wyner’s common information for this case.

An m-extracting procedure produces unbiased random bits from a loaded dice with m faces. A binarization takes inputs from an m-faced dice and produce bit sequences to be fed into a (binary) extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an m-extracting procedure via a binarization. An entropy- preserving binarization is to be called complete, and such a procedure has been proposed by Zhou and Bruck. We show that there exist complete binarizations in abundance as naturally arising from binary trees with m leaves. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments., 8 pages

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances, and we discuss decoding algorithms and correctable patterns of errors and erasures. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes, generalizing a recent construction of capacity-achieving rate-compatible polar codes. Applications of rate-compatible codes to data storage are considered. We design two-level rate-compatible codes based on Bose-Chaudhuri-Hocquenghem (BCH) and low-density parity-check (LDPC) codes which are two popular codes widely used in the data storage industry, and then we evaluate the performance of these codes in multi-level cell (MLC) flash memories. We also examine code performance on binary and $q$ -ary symmetric channels. Finally, we briefly discuss two variations of our main construction and their relative performance.

Consider a binary maximum distance separable (MDS) array code composed of an $m\times (k+r)$ array of bits with $k$ information columns and $r$ parity columns, such that any $k$ out of $k+r$ columns suffice to reconstruct the $k$ information columns. Our goal is to provide optimal repair access for binary MDS array codes, meaning that the bandwidth triggered to repair any single failed information or parity column is minimized. In this paper, we propose a generic transformation framework for binary MDS array codes, using EVENODD codes as a motivating example, to support optimal repair access for $k+1\le d \le k+r-1$ , where $d$ denotes the number of non-failed columns that are connected for repair; note that when $d , some of the chosen $d$ columns in repairing a failed column are specific. In addition, we show how our transformation framework applies to an example of binary MDS array codes with asymptotically optimal repair access of any single information column and enables asymptotically or exactly optimal repair access for any column. Furthermore, we present a new transformation for EVENODD codes with two parity columns such that the existing efficient repair property of any information column is preserved and the repair access of parity column is optimal.

The $\mathbb {Z}_{2^{s}}$ -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 Hadamard code is a binary Hadamard code which is the Gray map image of a $\mathbb {Z}_{2^{s}}$ -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the $\mathbb {Z}_{4}$ -linear Hadamard codes. However, when $s > 2$ , the dimension of the kernel of $\mathbb {Z}_{2^{s}}$ -linear Hadamard codes of length $2^{t}$ only provides a complete classification for some values of $t$ and $s$ . In this paper, the rank of these codes is computed for $s=3$ . Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once $t\geq 3$ is fixed. In this case, the number of nonequivalent such codes is also established.

We present a simple framework for constructing $k$ -ary de Bruijn sequences, and more generally, universal cycles, via successor rules. The framework is based on the often used method of joining disjoint cycles. It generalizes several previously known de Bruijn sequence constructions based on the pure cycling register and is applied to derive a new construction that is perhaps the simplest of all successors. Furthermore, it generalizes an algorithm to construct binary de Bruijn sequences based on any arbitrary nonsingular feedback function. The framework is applied to derive and prove the correctness of successors to efficiently construct 1) universal cycles for $k$ -ary strings of length $n$ whose weight is bounded by some $w$ and 2) universal cycles for permutations. It has also been subsequently applied to find the first universal cycle constructions for weak orders.

A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region, called zero correlation zone (ZCZ). Optimal odd-length binary ZCPs (OB-ZCPs) display closest correlation properties to Golay complementary pairs (GCPs) in that each OB-ZCP achieves maximum ZCZ of width $({N}+1)/2$ (where N is the sequence length) and every out-of-zone AACSs reaches the minimum magnitude value, i.e. 2. Till date, systematic constructions of optimal OB-ZCPs exist only for lengths $2^{\alpha } \pm 1$ , where $\alpha $ is a positive integer. In this paper, we construct optimal OB-ZCPs of generic lengths $2^\alpha 10^\beta 26^\gamma +1$ (where $\alpha,~\beta,~\gamma $ are non-negative integers and $\alpha \geq 1$ ) from inserted versions of binary GCPs. The key leading to the proposed constructions is several newly identified structure properties of binary GCPs obtained from Turyn’s method. This key also allows us to construct OB-ZCPs with possible ZCZ widths of $4 \times 10^{\beta -1} +1$ , $12 \times 26^{\gamma -1}+1$ and $12 \times 10^\beta 26^{\gamma -1}+1$ through proper insertions of GCPs of lengths $10^\beta,~26^\gamma, \text {and } 10^\beta 26^\gamma $ , respectively. Our proposed OB-ZCPs have applications in communications and radar (as an alternative to GCPs).

The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between $2^{n}$ -periodic binary sequences and Boolean functions on $n$ variables, it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding Boolean function $f$ - namely, it yields an upper bound on the minimum Hamming distance between $f$ and the set of functions depending on fewer number of variables. Therefore, the prominent Lauder-Paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a Boolean function that depend on fewer number of variables. Moreover, it is also shown that, through this approach, low-degree approximations of a Boolean function can be also obtained in an efficient way.

We study natural examples of binary channels with synchronization errors. These include the duplication channel, which independently outputs a given bit once or twice, and geometric channels that repeat a given bit according to a geometric rule, with or without the possibility of bit deletion. We apply the general framework of Cheraghchi (STOC 2018) to obtain sharp analytical upper bounds on the capacity of these channels. Previously, upper bounds were known via numerical computations involving the computation of finite approximations of the channels by a computer and then using the obtained numerical results to upper bound the actual capacity. While leading to sharp numerical results, further progress on the full understanding of the channel capacity inherently remains elusive using such methods. Our results can be regarded as a major step towards a complete understanding of the capacity curves. Quantitatively, our upper bounds sharply approach, and in some cases surpass, the bounds that were previously only known by purely numerical methods. Among our results, we notably give a completely analytical proof that, when the number of repetitions per bit is geometric (supported on $\{0,1,2,\dots\}$) with mean growing to infinity, the channel capacity remains substantially bounded away from $1$., Comment: 37 pages, 12 figures. Fixed some typos and reorganized parts of Section 5

Binary maximum distance separable (MDS) array codes are a special class of erasure codes for distributed storage that not only provides fault tolerance with minimum storage redundancy but also achieves low computational complexity. They are constructed by encoding $k$ information columns into $r$ parity columns, in which each element in a column is a bit, such that any $k$ out of the $k+r$ columns suffice to recover all information bits. In addition to providing fault tolerance, it is critical to improve repair performance in practical applications. Specifically, if a single column fails, then our goal is to minimize the repair bandwidth by downloading the least amount of bits from $d$ healthy columns, where $k\leq d\leq k+r-1$ . If one column of an MDS code is failed, it is known that we need to download at least $1/(d-k+1)$ fraction of the data stored in each of the $d$ healthy columns. If this lower bound is achieved for the repair of the failure column from accessing arbitrary $d$ healthy columns, we say that the MDS code has optimal repair. However, if such lower bound is only achieved by $d$ specific healthy columns, then we say the MDS code has weak-optimal repair. In this paper, we propose two explicit constructions of binary MDS array codes with more parity columns (i.e., $r\geq 3$ ) that achieve asymptotically weak-optimal repair, where $k+1\leq d\leq k+\lfloor (r-1)/2\rfloor $ , and “asymptotic” means that the repair bandwidth achieves the minimum value asymptotically in $d$ . Codes in the first construction have odd number of parity columns and asymptotically weak-optimal repair for anyone information failure, while codes in the second construction have even number of parity columns and asymptotically weak-optimal repair for any one column failure.

In this paper, we establish a new bound tying together the effective length and the maximum correlation between the outputs of an arbitrary pair of Boolean functions which operate on two sequences of correlated random variables. We derive a new upper bound on the correlation between the outputs of these functions. The upper bound may find applications in problems in many areas which deal with common information. We build upon Witsenhausen’s result [1] on maximum correlation. The present upper bound takes into account the effective length of the Boolean functions in characterizing the correlation. We use the new bound to characterize the communication-cooperation tradeoff in multi-terminal communications. We investigate binary block-codes (BBC). A BBC is defined as a vector of Boolean functions. We consider an ensemble of BBCs which is randomly generated using single-letter distributions. We characterize the vector of dependency spectrums of these BBCs. We use this vector to bound the correlation between the outputs of two distributed BBCs. Finally, the upper bound is used to show that the large blocklength single-letter coding schemes studied in the literature are sub-optimal in various multi-terminal communication settings.

We construct error-correcting write-once memory (WOM) codes that guarantee correction of any specified number of errors in $q$ -level memories. The constructions use suitably designed short $q$ -ary WOM codes and concatenate them with outer error-correcting codes over different alphabets using suitably designed mappings. With a new storage-efficiency measure, we call EC-rate and show that for common error types the codes save redundancy and implementation complexity over straightforward concatenation. In addition to constructions for guaranteed error correction, we extend the error-correcting WOM scheme to binary multi-level coding for random errors, and toward soft-decision decoding provide an efficient way to extract reliability information without using higher-precision readout.

It was shown in \cite{GXY18} that the length $n$ of a $q$-ary linear locally recoverable code with distance $d\ge 5$ is upper bounded by $O(dq^3)$. Thus, it is a challenging problem to construct $q$-ary locally recoverable codes with distance $d\ge 5$ and length approaching the upper bound. The paper \cite{GXY18} also gave an algorithmic construction of $q$-ary locally recoverable codes with locality $r$ and length $n=\Omega_r(q^2)$ for $d=5$ and $6$, where $\Omega_r$ means that the implicit constant depends on locality $r$. In the present paper, we present an explicit construction of $q$-ary locally recoverable codes of distance $d= 5$ and $6$ via binary constant weight codes. It turns out that (i) our construction is simpler and more explicit; and (ii) lengths of our codes are larger than those given in \cite{GXY18}., Comment: arXiv admin note: text overlap with arXiv:1807.01064 by other authors

In this paper, we study the close relationship between Reed-Muller codes and single-qubit phase gates from the perspective of $T$-count optimization. We prove that minimizing the number of $T$ gates in an $n$-qubit quantum circuit over CNOT and $T$, together with the Clifford group powers of $T$, corresponds to finding a minimum distance decoding of a length $2^n-1$ binary vector in the order $n-4$ punctured Reed-Muller code. Moreover, we show that the problems are polynomially equivalent in the length of the code. As a consequence, we derive an algorithm for the optimization of $T$-count in quantum circuits based on Reed-Muller decoders, along with a new upper bound of $O(n^2)$ on the number of $T$ gates required to implement an $n$-qubit unitary over CNOT and $T$ gates. We further generalize this result to show that minimizing small angle rotations corresponds to decoding lower order binary Reed-Muller codes. In particular, we show that minimizing the number of $R_Z(2\pi/d)$ gates for any integer $d$ is equivalent to minimum distance decoding in $\mathcal{RM}(n - k - 1, n)^*$, where $k$ is the highest power of $2$ dividing $d$., Comment: 19 pages. Version 2 gives a substantially different presentation of the results, as well as a generalization to rotation angles of any finite order

For locally repairable codes (LRCs), Cadambe and Mazumdar derived the first field-dependent parameter bound, known as the C-M bound. However, the C-M bound depends on an undetermined parameter $ {k}^{( {q})}_{{\textbf {opt}}}( {n}, {d})$ . In this paper, a sphere-packing approach is developed for upper bounding the parameter $ {k}$ for $[ {n}, {k}, {d}]$ linear LRCs with locality $ {r}$ . When restricted to the binary field, three upper bounds (i.e., Bound A , Bound B , and Bound C ) are derived in an explicit form. More specifically, Bound A holds under the hypothesis that the local repair groups are disjoint and of equal size. Comparing with previous bounds obtained under the same hypothesis, Bound A either covers them as special cases or has an advantage due to its explicit form. Then, the hypothesis is removed in Bound B and Bound C . As the price for explicit form, Bound B specially holds for $ {d}\geq \text {5}$ and Bound C for $ {r}=\textbf {2}$ . Through specific comparisons, we show that Bound B and Bound C both tend to outperform the C-M bound, as $ {n}$ goes large. Moreover, a family of binary linear LRCs with $ {d}\geq \textbf {6}$ attaining Bound B are constructed and later extended to a wider range of parameters by a shortening technique. Lastly, most of the bounds and constructions are extended to $ {q}$ -ary LRCs.

Lattice codes are elegant and powerful structures that not only can achieve the capacity of the 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 LDPC lattices constructed via Construction D' whose complexity is linear in the total number of coded bits. Moreover, a generalization of Construction D' is proposed that 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 (LDLC) on the power-unconstrained AWGN channel., 15 pages, 4 figures. To appear in IEEE Transactions on Information Theory

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed, and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed–Muller codes serve as an important and explicit class of examples.

In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$., arXiv admin note: substantial text overlap with arXiv:1612.06335. The published version of this paper incorrectly states the alphabet size in Theorem 3.4. This version states the result correctly

Marton’s inner bound is the best known achievable rate region for a general two-receiver discrete memoryless broadcast channel. In this paper, we establish improved bounds on the cardinalities of the auxiliary random variables appearing in this inner bound to the true rate region. We combine a perturbation technique, along with a representation using concave envelopes of information-theoretic functions that involve the use of auxiliary random variables, to achieve this improvement. The new cardinality bounds lead to a proof that a randomized-time-division strategy achieves every rate triple in Marton’s region for binary input broadcast channels. This extends the result by Hajek and Pursley which showed that the Cover-van der Muelen region was exhausted by the randomized-time-division strategy.

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained.

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.

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.