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. In this paper, we further determine the complete weight hierarchy of linear codes from a quadratic form over a finite field of even characteristic by carefully studying the behavior of the quadratic form on the subspaces of this field and its dual space, and complement the results of Li and Li.
A weighted Hamming metric is introduced in [4] and it showed that the binary generalized Goppa code is a perfect code in some weighted Hamming metric. In this paper, we study the weight structures which admit the binary Hamming code and the extended binary Hamming code to be perfect codes in the weighted Hamming metric. And, we also give some structures of a 2-perfect code and how to construct a 2-perfect code in some weighted Hamming metric.
The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. In this paper, we investigate the generalized Hamming weights of two classes of linear codes constructed from defining sets and determine them completely employing a number-theoretic approach.
We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative generalized Hamming weights of Reed-Muller codes by Geil and Martin, as well as the determination of all the generalized Hamming weights of the affine Cartesian codes by Beelen and Datta.
In this note, we give a very simple description of the generalized Hamming weights of Reed--Muller codes. For this purpose, we generalize the well-known Macaulay representation of a nonnegative integer and state some of its basic properties.
The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve an problem left in Yang et al. (IEEE Trans. Inf. Theory 61(9): 4905--4913, 2015).
Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both cases have been established and proven in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the ran…
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.
We define generalized Hamming weights for almost affine codes. We show how various aspects and applications of generalized Hamming weights for linear codes, such as Wei duality, generalized Kung's bound, profiles, connection to wire-tap channels of type II, apply to the larger class of almost affine codes in general. In addition we discuss duality of almost affine codes,and of the smaller class of multilinear codes. We also give results about weight distributions of infinite series of almost affine codes, each series obtained from a fixed code by extending the code alphabet.
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…
Firstly, we give a formula on the generalized Hamming weight of linear codes constructed generically by defining sets. Secondly, by choosing properly the defining set we obtain a class of cyclotomic linear codes and then present two alternative formulas to calculate their generalized Hamming weights. Lastly, we determine their weight distribution and generalized Hamming weights partially. Especially, we solved the generalized Hamming weights completely in one case.
Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made. The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes.
The Hamming weight enumerator function of the formally self-dual even, binary extended quadratic residue code of prime p = 8m + 1 is given by Gleason's theorem for singly-even code. Using this theorem, the Hamming weight distribution of the extended quadratic residue is completely determined once the number of codewords of Hamming weight j A_j, for 0 <= j <= 2m, are known. The smallest prime for which the Hamming weight distribution of the corresponding extended quadratic residue code is unknown is 137. It is shown in this paper that, for p=137 A_2m = A_34 may be obtained with out the need of …
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）
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes. This is further developed for Hermitian codes and the codes on one of the Garcia-Stichtenoth towers, as well as for some more general families.
Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric is a generalization for Hamming metric. The complete $b$-symbol Hamming weight distribution of irreducible codes is known in only a few cases. In this paper, we give a complete $b$-symbol Hamming weight distribution of a class of irreducible codes with two nonzero $b$-symbol Hamming weights.
Coset constructions of $q$-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local error-correction. In this paper we determine the relative generalized Hamming weights of these codes which can be translated into a detailed description of the information leakage [2, 24, 21, 11].
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 $m \geq 3$.
Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights.
In this paper, we characterize the average Hamming weight distribution of subsequences of maximum-length sequences ($m$-sequences). In particular, we consider all possible $m$-sequences of dimension $k$ and find the average number of subsequences of length $n$ that have a Hamming weight $t$. To do so, we first characterize the Hamming weight distribution of the average dual code and use the MacWilliams identity to find the average Hamming weight distribution of subsequences of $m$-sequences. We further find a lower bound on the minimum Hamming weight of the subsequences and show that there alw…
We derive a recursive formula determing the weight distribution of the [n=(q^m-1)/(q-1), n-m, 3] Hamming code H(m,q), when (m, q-1)=1. Here q is a prime power. The proof is based on Moisio's idea of using Pless power moment identity together with exponential sum techniques.
Relative generalized Hamming weights (RGHWs) of a linear code respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the code. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes with two nonzeros respect to any of its irreducible cyclic subcodes. Applying the method in the paper [arxiv.org/abs/1410.2702], we give two for…
In a previous paper, we derived a recursive formula determining the weight distributions of the [n=(q^m-1)/(q-1)] Hamming code H(m,q), when (m,q-1)=1. Here q is a prime power. We note here that the formula actually holds for any positive integer m and any prime power q, without the restriction (m, q-1)=1.
We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q. These codes appeared recently in a work by H. Lopez, C. Renteria-Marquez and R. Villareal and, in a generalized form, in a work by O. Geil and C. Thomsen. Using methods from Gröbner basis theory we determine the second Hamming weight (also called next-to-minimal weight) for particular cases of affine cartesian codes and also some higher Hamming weights of this type of code.
In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.
In this paper, we discuss the generalized Hamming weights of a class of linear codes associated with non-degenerate quadratic forms. In order to do so, we study the quadratic forms over subspaces of finite field and obtain some interesting results about subspaces and their dual spaces. On this basis, we solve all the generalized Hamming weights of these linear codes.
Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length $p^s$, where $p$ is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a chain ring. In this paper, all remaining cases of cyclic and negacyclic codes of length $p^s$ over a Galois ring alphabet are considered. In these cases the ambient space is a local ring with simpl…
We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015.
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify for each dimension $ν$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $ν$. The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work by the same authors, a code-shortening algori…
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…
There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized Hamming weight for a family of affine cartesian codes. If $\mathbb{X}$ is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on $\mathbb{X}$ and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed--…
We consider linear codes over some fixed finite field extension over an arbitrary finite field. Gabidulin introduced rank metric codes, by endowing linear codes over the extension field with a rank weight over the base field and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of wiretap II codes in terms of generalized Hamming weights by Wei, Kurihara et al. defined some generalized rank weights and showed their relevance for secure network coding. In this paper, we derive a statement for generalized rank weights …
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.
It was recently shown that RGHW (relative generalized Hamming weight) exactly expresses the security of linear ramp secret sharing scheme. In this paper we determine the true value of the asymptotic metric for RGHW previously proposed by Zhuang et al. in 2013. Then we propose new asymptotic metrics useful for investigating the optimal performance of linear ramp secret sharing scheme constructed from a pair of linear codes. We also determine the true values of the proposed metrics in many cases.
Conflict-avoiding codes are used in the multiple-access collision channel without feedback. The number of codewords in a conflict-avoiding code is the number of potential users that can be supported in the system. In this paper, a new upper bound on the size of conflict-avoiding codes is proved. This upper bound is general in the sense that it is applicable to all code lengths and all Hamming weights. Several existing constructions for conflict-avoiding codes, which are known to be optimal for Hamming weights equal to four and five, are shown to be optimal for all Hamming weights in general.
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）
Binary constant-weight codes have been extensively studied, due to both their numerous applications and to their theoretical significance. In particular, constant-weight codes have been proposed for error correction in store and forward. In this paper, we introduce constant-weight array codes (CWACs), which offer a tradeoff between the rate gain of general constant-weight codes and the low decoding complexity of liftings. CWACs can either be used in the on-shot setting introduced earlier or in a multi-shot approach, where one codeword consists of several messages. The multi-shot approach gener…
Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-type code over the set $\mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${\rm char}(K)\neq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity o…
Past few years have seen an extensive use of RAM chips with wide I/O data (e.g. 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an $m$-spotty byte error, which can be effectively detected or corrected using byte error-control codes. The MacWilliams identity provides the relationship between the weight distribution of a code and that of its dual. This paper introduces $m$-spotty Hamming weight enumerator, joint $m$-spotty Hamming weight enumerator and split $m$-spotty Hamming weight enumerator for byte error-control codes ov…
Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over $\mathbb{F}_{q}$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$ of any set of $r$ linearly independent square-free polynomials over $\mathbb{F}_{q}$ in $s$ variables, under certain conditions on $r$, $s$ and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluatio…
A new protograph-based framework for message passing (MP) decoding of low density parity-check (LDPC) codes with Hamming weight amplifiers (HWAs), which are used e.g. in the NIST post-quantum crypto candidate LEDAcrypt, is proposed. The scheme exploits the correlations in the error patterns introduced by the HWA using a turbo-like decoding approach where messages between the decoders for the outer code given by the HWA and the inner LDPC code are exchanged. Decoding thresholds for the proposed scheme are computed using density evolution (DE) analysis for belief propagation (BP) and ternary mes…
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.
Let $R$ be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over $R$ and its dual hold, if and only if, $R$ is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of $R$ are the same, if and only if, for every linear code $C$ over $R$, the dual of the dual $C$ is $C$ itself. Also as an intermediate step, we present a new and simpler proof for the commutative case of Wood's theorem which states that $R$ has a generating character if and only if $R$ is a Frob…
