Jean Creignou, Hervé Diet
The linear programming method is applied to the space $\U_n(\C)$ of unitary matrices in order to obtain bounds for codes relative to the diversity sum and the diversity product. Theoretical and numerical results improving previously known bounds are derived.
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We provide an algorithm to construct unitary matrices over finite fields. We present various constructions of Hermitian self-dual code by means of unitary matrices, where some of them generalize the quadratic double circulant constructions. Many optimal Hermitian self-dual codes over large finite fields with new parameters are obtained. More precisely MDS or almost MDS Hermitian self-dual codes of lengths up to $18$ are constructed over finite fields $\F_{q},$ where $q=3^2,4^2,5^2,7^2,8^2,9^2,11^2,13^2,17^2,19^2.$ Comparisons with classical constructions are made.
We apply the semidefinite programming method to derive bounds for projective codes over a finite field.
It is well known that the Space-time Block Codes (STBCs) from Complex orthogonal designs (CODs) are single-symbol decodable/symbol-by-symbol decodable (SSD). The weight matrices of the square CODs are all unitary and obtainable from the unitary matrix representations of Clifford Algebras when the number of transmit antennas $n$ is a power of 2. The rate of the square CODs for $n = 2^a$ has been shown to be $\frac{a+1}{2^a}$ complex symbols per channel use. However, SSD codes having unitary-weight matrices need not be CODs, an example being the Minimum-Decoding-Complexity STBCs from Quasi-Ortho…
Finding the largest code with a given minimum distance is one of the most basic problems in coding theory. In this paper, we study the linear programming bound for codes in the Lee metric. We introduce refinements on the linear programming bound for linear codes in the Lee metric, which give a tighter bound for linear codes. We also discuss the computational aspects of the problem and introduce a more compact program by the obtained refinements, and a recursive method for computing the so-called Lee-numbers, which are important in the computation of the linear programming bound.
Let $A(n,d,w)$ be the largest possible size of an $(n,d,w)$ constant-weight binary code. By adding new constraints to Delsarte linear programming, we obtain twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$. The used techniques allow us to give a simple proof of an important theorem of Delsarte which makes linear programming possible for binary codes.
In previous work, we demonstrated how decoding of a non-binary linear code could be formulated as a linear-programming problem. In this paper, we study different polytopes for use with linear-programming decoding, and show that for many classes of codes these polytopes yield a complexity advantage for decoding. These representations lead to polynomial-time decoders for a wide variety of classical non-binary linear codes.
Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.
Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.
It is well known that Space-Time Block Codes (STBCs) from orthogonal designs (ODs) are single-symbol decodable/symbol-by-symbol decodable (SSD) and are obtainable from unitary matrix representations of Clifford algebras. However, SSD codes are obtainable from designs that are not orthogonal also. Recently, two such classes of SSD codes have been studied: (i) Coordinate Interleaved Orthogonal Designs (CIODs) and (ii) Minimum-Decoding-Complexity (MDC) STBCs from Quasi-ODs (QODs). Codes from ODs, CIODs and MDC-QODs are mutually non-intersecting classes of codes. The class of CIODs have {\it non-u…
A coded modulation system is considered in which nonbinary coded symbols are mapped directly to nonbinary modulation signals. It is proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. It is shown that this result holds for both linear-programming decoders and sum-product decoders. In particular, this provides a natural modulation mapping for nonbinary codes mapped to PSK constellations for transmission over memoryless channels such as AWGN channels or flat fading channels wit…
We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA(2) codes.
We develop two new designs of unitary differential space-time modulation (DSTM) with low decoding complexity. Their decoder can be separated into a few parallel decoders, each of which has a decoding search space of less than sqrt(N) if the DSTM codebook contains N codewords. Both designs are based on the concept of joint modulation, which means that several information symbols are jointly modulated, unlike the conventional symbol-by-symbol modulation. The first design is based on Orthogonal Space-Time Block Code (O-STBC) with joint constellation constructed from spherical codes. The second de…
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices)…
In this paper we present a new Turbo analog error correcting coding scheme for real valued signals that are corrupted by impulsive noise. This Turbo code improves Donoho's deterministic construction by using a probabilistic approach. More specifically, our construction corrects more errors than the matrices of Donoho by allowing a vanishingly small probability of error (with the increase in block size). The problem of decoding the long block code is decoupled into two sets of parallel Linear Programming problems. This leads to a significant reduction in decoding complexity as compared to one-s…
Cyclic codes over a commutative non-unitary ring of order 4
研究4阶交换非酉环\(I_2\)上的循环码，通过相关剩余码和挠码，引入扭曲与非扭曲循环码概念，利用格雷映射建立与二元准循环码联系，研究对偶性质，对长度\(n\leq7\)的码分类并确定其结构性质。
A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 δ_A δ_B (where δ_A and δ_B are the relative minimum distances of the constituent codes).
Because of the emerging field of Internet of Things (IoT), future backscatter RFID is required to be more reliable and data intensive. Motivated by this, orthogonal space-time block code (OSTBC), which is very successful in mobile communications for its low complexity and high performance, has already been investigated for backscatter RFID. On the other hand, a recently proposed scheme called unitary query was shown to be able to considerably improve the reliability of backscatter radio by exploiting query diversity. Therefore incorporating the classical OSTBC (at the tag end) with the recentl…
In this paper we propose a new design criterion and a new class of unitary signal constellations for differential space-time modulation for multiple-antenna systems over Rayleigh flat-fading channels with unknown fading coefficients. Extensive simulations show that the new codes have significantly better performance than existing codes. We have compared the performance of our codes with differential detection schemes using orthogonal design, Cayley differential codes, fixed-point-free group codes and product of groups and for the same bit error rate, our codes allow smaller signal to noise rat…
In this paper, an integer programming approach is introduced to construct Unequal Error Protection (UEP) codes for multiuser broadcast channels. We show that the optimal codes can be constructed that satisfy the integer programming bound. Based on the bound, we compute asymptotic code rate and perform throughput analysis for the degraded broadcast channel.
We consider a point-to-point flat-fading MIMO channel with channel state information known both at transmitter and receiver. At the transmitter side, a lattice coding scheme is employed at each antenna to map information symbols to independent lattice codewords drawn from the same codebook. Each lattice codeword is then multiplied by a unitary precoding matrix ${\bf P}$ and sent through the channel. At the receiver side, an integer-forcing (IF) linear receiver is employed. We denote this scheme as unitary precoded integer-forcing (UPIF). We show that UPIF can achieve full-diversity under a con…
We propose several improvements for Linear Programming (LP) decoding algorithms for High Density Parity Check (HDPC) codes. First, we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks. Second, we propose an efficient mixed integer decoder utilizing the branch and bound method. We further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations. Based on simulation results the proposed decoders achieve near-ML p…
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the 'maximum-likelihood certificate' property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particul…
We examine LDPC codes decoded using linear programming (LP). Four contributions to the LP framework are presented. First, a new method of tightening the LP relaxation, and thus improving the LP decoder, is proposed. Second, we present an algorithm which calculates a lower bound on the minimum distance of a specific code. This algorithm exhibits complexity which scales quadratically with the block length. Third, we propose a method to obtain a tight lower bound on the fractional distance, also with quadratic complexity, and thus less than previously-existing methods. Finally, we show how the fu…
We develop a framework for linear-programming (LP) decoding of non-binary linear codes over rings. We prove that the resulting LP decoder has the `maximum likelihood certificate' property, and we show that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. LP decoding performance is illustrated for the (11,6,5) ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the LP decoder performance is comparable to codeword-error-rate-optimum hard-decision b…
On the difficulty to beat the first linear programming bound for binary codes
本文研究二元码首个线性规划界，指出基于Friedman-Tillich工作的较新论证方法在试图超越该界（如迈向第二个线性规划界）时会遇到困难。
We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest w…
In this paper we explore the decision regions of Linear Programming (LP) decoding. We compare the decision regions of an LP decoder, a Belief Propagation (BP) decoder and the optimal Maximum Likelihood (ML) decoder. We study the effect of minimal-weight pseudocodewords on LP decoding. We present global optimization as a method for finding the minimal pseudoweight of a given code as well as the number of minimal-weight generators. We present a complete pseudoweight distribution for the [24; 12; 8] extended Golay code, and provide justifications of why the pseudoweight distribution alone cannot …
In this paper, a novel concept called a \textit{uniquely factorable constellation pair} (UFCP) is proposed for the systematic design of a noncoherent full diversity collaborative unitary space-time block code by normalizing two Alamouti codes for a wireless communication system having two transmitter antennas and a single receiver antenna. It is proved that such a unitary UFCP code assures the unique identification of both channel coefficients and transmitted signals in a noise-free case as well as full diversity for the noncoherent maximum likelihood (ML) receiver in a noise case. To further …
We revisit the linear programming bounds for the size vs. distance trade-off for binary codes, focusing on the bounds for the almost-balanced case, when all pairwise distances are between $d$ and $n-d$, where $d$ is the code distance and $n$ is the block length. We give an optimal solution to Delsarte's LP for the almost-balanced case with large distance $d \geq (n - \sqrt{n})/2 + 1$, which shows that the optimal value of the LP coincides with the Grey-Rankin bound for self-complementary codes. We also show that a limitation of the asymptotic LP bound shown by Samorodnitsky, namely that it is …
A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to have multiple repair groups. In this paper we consider robust locally repairable coding schemes which guarantee that there exist multiple distinct (not necessarily disjoint) alternative local repair groups for any single failure such that the failed node can still be repaired locally even if some of the repair groups are not available. We use linear programming…
In this paper, we develop a new decoding algorithm of a binary linear codes for symbol-pair read channels. Symbol-pair read channel has recently been introduced by Cassuto and Blaum to model channels with high write resolution but low read resolution. The proposed decoding algorithm is based on a linear programming (LP). It is proved that the proposed LP decoder has the maximum-likelihood (ML) certificate property, i.e., the output of the decoder is guaranteed to be the ML codeword when it is integral. We also introduce the fractional pair distance $d_{fp}$ of a code which is a lower bound on …
In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all in…
In this letter, we develop an efficient linear programming (LP) decoding algorithm for low-density parity-check (LDPC) codes. We first relax the maximum likelihood (ML) decoding problem to a LP problem by using check-node decomposition. Then, to solve the resulting LP problem, we propose an efficient iterative algorithm based on the alternating direction method of multipliers (ADMM) technique. In addition, the feasibility analysis of the proposed algorithm is presented. Furthermore, through exploiting the sparsity and orthogonality structures of the LP problem, the computational complexity of …
Construction of self-orthogonal codes over a commutative non-unitary ring of order 25
研究阶为25的交换非酉环\(I_5\)上的线性码，定义三类线性\(I_5\) - 码并研究其结构，对长度至多为4的三类码进行单项式等价完全分类，同时纠正了\(I_5\)上拟自对偶码分类的错误。
We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in the projective spaces.
Recent interest on permutation rank modulation shows the Kendall tau metric as an important distance metric. This note documents our first efforts to obtain upper bounds on optimal code sizes (for said metric) ala Delsarte's approach. For the Hamming metric, Delsarte's seminal work on powerful linear programming (LP) bounds have been extended to permutation codes, via association scheme theory. For the Kendall tau metric, the same extension needs the more general theory of coherent configurations, whereby the optimal code size problem can be formulated as an extremely huge semidefinite program…
The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters $n_1>n_2$, where $n_1 = \left\lceil \frac{n}{r+1} \right\rceil$ and $n_2 = n_1 (r+1) - n$. Finally, an explic…
In the standard polar code construction, the message vector $(U_0,U_1,\dots,U_{n-1})$ is divided into information bits and frozen bits according to the reliability of each $U_i$ given $(U_0,U_1,\dots,U_{i-1})$ and all the channel outputs. While this reliability function is the most suitable measure to choose information bits under the Successive Cancellation (SC) decoder, there is a mismatch between this reliability function and the Successive Cancellation List (SCL) decoder because the SCL decoder also makes use of the information from the future frozen bits. We propose a Dynamic Programming …
Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on $A(n,d)$, namely $A(18,8) \leq 71$ and $A(19,8) \leq 131$. Twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$ are also obtained by a similar way.
Computer classification of linear codes based on lattice point enumeration and integer linear programming
针对有限域上受限权重集线性码的分类问题，提出基于格点枚举和整数线性规划的算法框架，给出射影双权重码等三类码的新枚举及不存在性结果。
We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym$(n)$. In particular, we compute orbits of ordered pairs on Sym$(n)$ acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size $M(n,d)$ of permutation codes of lengths up to 7. For instance, these techniques detect the nonexistence of the projective plane of order six via $M(6,5)<30$ and yield a new best bound $M(7,4) \le 535$ for a challenging open case. Each of the…
Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators
本文利用分裂重量枚举器推导纠缠辅助量子码的线性规划界，改进了小型量子码最小距离界，证明两类量子稳定器码不存在，优化了短码的距离上下界。
We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance $δn$ is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to $\left(\frac 12 - \sqrt{δ(1-δ)}\right) \cdot n$ are essentially independent.…
The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organi…
Semidefinite programming bounds for binary codes from a split Terwilliger algebra
该研究针对二元编码最大规模$A(n,d)$的上界问题，基于分裂Terwilliger代数推导更精细的矩阵不等式，改进了Schrijver的半定规划界，并将$A(18,4)$的界提升至6551。
Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
本文将半定规划应用于纠缠辅助码字稳定化（CWS）量子码，利用同向子群和字算符集推导出最小距离上界，并构建SDP约束获得了比LP更紧的尺寸和距离界。
We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte's linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower …
In his Ph.D. disseration, Feldman and his collaborators define the linear programming decoder for binary linear codes, which is a linear programming relaxation of the maximum-likelihood decoding problem. This decoder does not, in general, attain maximum-likelihood performance; however, the source of this discrepancy is known to be the presence of non-integral extreme points (vertices) within the fundamental polytope, vectors which are also called nontrivial linear programming pseudocodewords. Restricting to the class of cycle codes, we provide necessary conditions for a vector to be a linear p…
This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Solé. This book will collect short but foundational articles, emphasizing definitions, examples, exhaustive references, and basic facts on the model of the Handbook of Finite Fields. The target audience of the Encyclopedia is upper level undergraduates and graduate students.
Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. The connection to matrix-variate hypergeometric functions and Szegő's strong limit theorem lead independently from the finite size formula to an asymptotic one. The convergence of the limiting formula is exceptionally fast due to…
Tamper Detection against Unitary Operators
本研究将篡改检测码扩展到量子对手场景，证明在酉算子族规模小于2^{2^{αn}}且不过于接近恒等算子的条件下，量子篡改检测码存在，并可为经典消息集实现非可延展码。
Error-Correcting Output Codes (ECOCs) offer a principled approach for combining simple binary classifiers into multiclass classifiers. In this paper, we investigate the problem of designing optimal ECOCs to achieve both nominal and adversarial accuracy using Support Vector Machines (SVMs) and binary deep learning models. In contrast to previous literature, we present an Integer Programming (IP) formulation to design minimal codebooks with desirable error correcting properties. Our work leverages the advances in IP solvers to generate codebooks with optimality guarantees. To achieve tractabilit…
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