In this letter we consider the ensemble of codes formed by the serial concatenation of a Hamming code and two accumulate codes. We show that this ensemble is asymptotically good, in the sense that most codes in the ensemble have minimum distance growing linearly with the block length. Thus, the resulting codes achieve high minimum distances with high probability, about half or more of the minimum distance of a typical random linear code of the same rate and length in our examples. The proposed codes also show reasonably good iterative convergence thresholds, which makes them attractive for applications requiring high code rates and low error rates, such as optical communications and magnetic recording.
We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + γ)^{\ell}$ and $(x^n - ξ)^i(x^n + ξ)^j$\ respectively, where $x^n + γ$, $x^n - ξ$ and $x^n + ξ$ are irreducible over the alphabet $\F_{p^a}$. We generalize the results of [5], [6] and [7] by computing the minimum Hamming distance of these codes. As a particular case, we determine the minimum Hamming distance of cyclic and negacyclic codes, of length $2p^s$, over a finite field of characteristic $p$.
In this paper, we describe the properties of the $i$-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every $q$-ary code of length $m$ and minimum distance 5 (for $q = 3$ the minimum distance is 3) can be embedded in a $q$-ary 1-perfect code of length $n = (q^{m}-1)/(q-1)$. It is also shown that every binary code of length $m + k$ and minimum distance $3k + 3$ can be embedded in a binary 1-perfect code of length $n = 2^{m}-1$.
In this paper, we introduce generalized extended Hamming codes over Galois rings $GR(2^n,m)$ of characteristic $2^n$ with extension degree $m$. Furthermore we prove that the minimum Hamming weight of generalized extended Hamming codes over $GR(2^n,m)$ is 4 and the minimum Lee weight of generalized extended Hamming codes over $GR(8,m)$ is 6 for all $m \geq 3$.
In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds. Because of the tight algebraic connection b…
Polar code, with explicit construction and recursive structure, is the latest breakthrough in channel coding field for its low-complexity and theoretically capacity-achieving property. Since polar codes can approach the maximum likelihood performance under successive cancellation list decoding (SCLD), its decoding performance can be evaluated by Bonferroni-type bounds (e.g., union bound) in which the Hamming weight spectrum will be used. Especially, the polar codewords with minimum Hamming weight (PC-MHW) are the most important item in that bound because they make major contributions to the de…
This paper presents a new decoding for polynomial residue codes, called the minimum degree-weighted distance decoding. The newly proposed decoding is based on the degree-weighted distance and different from the traditional minimum Hamming distance decoding. It is shown that for the two types of minimum distance decoders, i.e., the minimum degree-weighted distance decoding and the minimum Hamming distance decoding, one is not absolutely stronger than the other, but they can complement each other from different points of view.
Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least $ 3 $ (thus one-error-correcting) for a fixed redundancy $ r $, base-field size $ q $ and field-extension degree $ m $ (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is $ 1 $, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective corresponde…
The generalized Hamming weight (GHW) $d_r(C)$ of linear codes $C$ is a natural generalization of the minimum Hamming distance $d(C)(=d_1(C))$ and has become one of important research objects in coding theory since Wei's originary work [23] in 1991. In this paper two general formulas on $d_r(C)$ for irreducible cyclic codes are presented by using Gauss sums and the weight hierarchy $\{d_1(C), d_2(C), \ldots, d_k(C)\}$ $(k=\dim C)$ are completely determined for several cases.
Generalized Hamming Weights and Subcode Support Weight Distributions of Twisted Reed-Solomon Codes
College of Information Science and Technology, Jinan University（暨南大学信息科学技术学院）；Hubei Laboratory of Applied Mathematics, Faculty of Mathematics, Hubei University（湖北大学数学学院应用数学实验室）；College of Cyber Security, Jinan University（暨南大学网络空间安全学院）
We discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo, d/n=delta. The problem is relevant for coding theory. We find a solution to the equations describing the liquid up to very large values of the density, but we show that this solution gives a negative entropy for the liquid phase when the density is large enough. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition. Finally we discuss the relation between our…
Let G be a finite strongly connected aperiodic directed graph in which each edge carries a label from a finite alphabet A. Then G induces a trellis coded quantizer for encoding an alphabet A memoryless source. A source sequence of long finite length is encoded by finding a path in G of that length whose sequence of labels is closest in Hamming distance to the source sequence; finding the minimum distance path is a dynamic programming problem that is solved using the Viterbi algorithm. We show how a Markov chain can be used to obtain a closed form expression for the asymptotic expected Hamming …
A pair $(T_0,T_1)$ of disjoint sets of vertices of a graph $G$ is called a perfect bitrade in $G$ if any ball of radius 1 in $G$ contains exactly one vertex in $T_0$ and $T_1$ or none simultaneously. The volume of a perfect bitrade $(T_0,T_1)$ is the size of $T_0$. In particular, if $C_0$ and $C_1$ are distinct perfect codes with minimum distance $3$ in $G$ then $(C_0\setminus C_1,C_1\setminus C_0)$ is a perfect bitrade. For any $q\geq 3$, $r\geq 1$ we construct perfect bitrades in the Hamming graph $H(qr+1,q)$ of volume $(q!)^r$ and show that for $r=1$ their volume is minimum.
Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper c…
Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall's $τ$-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert-Varshamov bound asymptotically by a factor $n$, when the minimum …
The multicast capacity of a directed network is closely related to the $s$-$t$ maximum flow, which is equal to the $s$-$t$ minimum cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or have stochastic nature, it is not so trivial to predict statistical properties on the maximum flow. In this paper, we present a coding theoretic approach for evaluating the accumulate distribution of the minimum cut capacity of weighted random graphs. The main feature of our approach is to utilize the correspondence between the cut space of…
For any prime $p$, $λ$-constacyclic codes of length $p^s$ over ${\cal R}=\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ are precisely the ideals of the local ring ${\cal R}_λ=\frac{{\cal R}[x]}{\left\langle x^{p^s}-λ\right\rangle}$, where $u^2=0$. In this paper, we first investigate the Hamming distances of cyclic codes of length $p^s$ over ${\cal R}$. The minimum Hamming distances of all cyclic codes of length $p^s$ over ${\cal R}$ are determined. Moreover, an isometry between cyclic and $α$-constacyclic codes of length $p^s$ over ${\cal R}$ is established, where $α$ is a nonzero element of $\mathbb{F…
The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which …
The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (IEEE Trans. Inf. Theory, 67(1): 124-129, 2021) and Li and Li (Discrete Math., 345: 112718, 2022) obtained the complete weight hierarchy of linear codes from a quadratic form over a finite field of odd characteristic by analysis of the solutions of the restricted quadratic equation in its subspace. I…
A new lower bound on the minimum Hamming distance of linear quasi-cyclic codes over finite fields is proposed. It is based on spectral analysis and generalizes the Semenov- Trifonov bound in a similar way as the Hartmann-Tzeng bound extends the BCH approach for cyclic codes. Furthermore, a syndrome-based algebraic decoding algorithm is given.
Optimal codes and arcs for the generalized Hamming weights
Universität Bayreuth（拜罗伊特大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；New Bulgarian University（新保加利亚大学）；Sofia University, Faculty of Mathematics and Informatics（索非亚大学数学与信息学院）
The Hamming oracle returns the Hamming distance between an unknown binary $n$-vector $x$ and a binary query $n$-vector y. The objective is to determine $x$ uniquely using a sequence of $m$ queries. What are the minimum number of queries required in the worst case? We consider the query ratio $m/n$ to be our figure of merit and derive upper bounds on the query ratio by explicitly constructing $(m,n)$ query matrices. We show that our recursive and algebraic construction results in query ratios arbitrarily close to zero. Our construction is based on codes of constant weight. A decoding algorithm …
We propose an algorithm using the Gaussian elimination method to find the minimal Hamming distance and decode received messages of linear codes. This algorithm is easy to implement as it requires no Gröbner bases to compute solutions for systems of polynomial equations
In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new Singleton type upper bounds on the sizes of codes with a given minimum Hamming distance. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes or list-decodable codes attaining the generalized Singleton bound are given. As an application of our general…
A new recursive construction of $N$-ary error-correcting output code (ECOC) matrices for ensemble classification methods is presented, generalizing the classic doubling construction for binary Hadamard matrices. Given any prime integer $N$, this deterministic construction generates base-$N$ symmetric square matrices $M$ of prime-power dimension having optimal minimum Hamming distance between any two of its rows and columns. Experimental results for six datasets demonstrate that using these deterministic coding matrices for $N$-ary ECOC classification yields comparable and in many cases higher …
In this paper we find the second generalized Hamming weight of some evaluation codes arising from a projective torus, and it allows to compute the second generalized Hamming weight of the codes parameterized by the edges of any complete bipartite graph. Also, at the beginning, we obtain some results about the generalized Hamming weights of some evaluation codes arising from a complete intersection when the minimum distance is known and they are non--degenerate codes. Finally we give an example where we use these results to determine the complete weight hierarchy of some codes.
In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan \textit{et al}, in…
In this paper the ensemble of codes formed by a serial concatenation of a repetition code with multiple accumulators connected through random interleavers is considered. Based on finite length weight enumerators for these codes, asymptotic expressions for the minimum distance and an arbitrary number of accumulators larger than one are derived using the uniform interleaver approach. In accordance with earlier results in the literature, it is first shown that the minimum distance of repeat-accumulate codes can grow, at best, sublinearly with block length. Then, for repeat-accumulate-accumulate c…
The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over Fp or Fp2 ; or over even degree extensions of Fp: In the latter case the approach is concatena- tion with a good code for the Hamming metric as outer code and a short code for the Lee metric as an inner code. In the former case lower bounds on the minimum Lee distance are derived by algebraic geometric arguments inspired by results of Wu, Kuijper, Udaya (2007).
We define the AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance. We show that most codes do not have a finite pseudocodeword redundancy. We also provide bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs.
We present an algorithm that, given a channel, determines if there is a distance for it such that the maximum likelihood decoder coincides with the minimum distance decoder. We also show that any metric, up to a decoding equivalence, can be isometrically embedded into the hypercube with the Hamming metric, and thus, in terms of decoding, the Hamming metric is universal.
Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower bound and an upper bound on $N(d,d^\perp)$. Further, for small values of $d$ and $d^\perp$, we determine $N(d,d^\perp)$ and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.
The insertion-deletion codes were motivated to correct the synchronization errors. In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes, which are based on the generalized Hamming weights and the formation of minimum Hamming weight codewords. Our bounds are stronger than some previous known bounds. We apply these upper bounds to some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. Some strong upper bounds on the insdel distances of Reed-Muller codes with special coordinate-ordering are also gi…
In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti-Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized hamming weights of such codes.
Message-passing iterative decoders for low-density parity-check (LDPC) block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing iterative decoding to be worse than that predicted by the maximum-likelihood decoding union bound. In this paper we address the pseudo-codeword problem from the convolutional-code perspective. In particular, we compare the performance of LDPC convolutional codes with that of their ``wrapped'' quasi-cyclic block versions and we show that the minimu…
In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Recently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-…
Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If $\mathbb{X}$ is a set of projective points over a finite field and $I$ is its vanishing ideal, we show that the gmd function and the Vasconcelos function of $I$ are equal to the $r$-th generalized Hamming weight of the correspondin…
We propose a binary message passing decoding algorithm for product codes based on generalized minimum distance decoding (GMDD) of the component codes, where the last stage of the GMDD makes a decision based on the Hamming distance metric. The proposed algorithm closes half of the gap between conventional iterative bounded distance decoding (iBDD) and turbo product decoding based on the Chase--Pyndiah algorithm, at the expense of some increase in complexity. Furthermore, the proposed algorithm entails only a limited increase in data flow compared to iBDD.
Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint-Wilson's shift and Hartmann-Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include al…
The MacWilliams Extension Theorem states that each linear Hamming isometry of a linear code extends to a monomial map. In this paper an analogue of the extension theorem for linear codes over a module alphabet is observed. A geometric approach to the extendability of isometries is described. For a matrix module alphabet we found the minimum length of a code for which an unextendable Hamming isometry exists. We also proved an extension theorem for MDS codes over a module alphabet.
We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection metric. In sharp contrast with the behavior of linear block codes, we show that the typical non-linear code in the Hamming metric of cardinality $q^{n-d+1}$ is far from having minimum distance $d$, i.e., from being MDS. We also give more precise results about the asymptotic proportion of block codes with good distance properties within the set of codes having a ce…
