Generalized Hamming weights of linear codes (original) (raw)

Double error Correcting Codes with Improved Code Rates

Journal of Electrical Engineering

In [1] a new family of error detection codes called Weighted Sum Codes was proposed. In [2] it was noted, that these codes are equivalent to lengthened Reed Solomon Codes, and shortened versions of lengthened Reed Solomon codes respectively, constructed over GF(2^(h/2)). It was also shown that it is possible to use these codes for correction of one error in each codeword over GF(2^(h/2)). In [3] a class of modified Generalized Weighted Sum Codes for single error and conditionally double error correction were presented. In this paper we present a new family of double error – correcting codes with code distance dm = 5. Weight spectrum for [59,49,5] code constructed over GF(8) which is an example of the new codes was obtained by computer using its dual [4]. The code rate of the new codes are higher than the code rate of ordinary Reed Solomon codes constructed over the same �finite fi�eld.

Generalized Hamming weights of three classes of linear codes

Finite Fields and Their Applications, 2017

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. (2015) [30]. Keywords Linear codes • Generalized Hamming weights • Gauss periods 1 Introduction Throughout this paper let p be an odd prime and q = p m for some positive integer m. Let n 1 be a positive integer coprime to p and without loss of generality we assume that m is the least positive integer such that p m ≡ 1 (mod n 1). Denote by Fp (or Fq) the finite field with p (or q) elements. Let α be a fixed primitive element of Fq. Let N = q−1 n1 , N 1 = gcd(N, q−1 p−1), N 2 = lcm(N, q−1 p−1) and θ = α N. Let Tr denote the trace function from Fq to Fp. An [n, k, d] linear code C over Fp is a k-dimensional subspace of F n p with minimum distance d. We recall the definition of the generalized Hamming weights of a linear code [28]. Suppose that U is an r-dimensional subspace of C, the support of U is defined to be Supp(U) = ∪ x∈U Supp(x), where Supp(x) is the set of coordinates where x is nonzero, i.e., Supp(U) = {i : 1 ≤ i ≤ n, x i = 0 for some x = (x 1 , x 2 ,. .. , xn) ∈ U }. Definition 1 Let C be an [n, k, d] linear code over Fp. For 1 ≤ r ≤ k, dr(C) = min{|Supp(U)| : U ⊂ C, dim U = r} is called the r-th generalized Hamming weight (GHW) of C and {dr(C) : 1 ≤ r ≤ k} is called the weight hierarchy of C.

Weight distributions of cosets of two-error-correcting binary BCH codes, extended or not

IEEE Transactions on Information Theory, 1994

Let B be the binary two-error-correcting BCH code of length 2"-1 and let B be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes B only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. In a recent paper, Camion, Courteau, and Montpetit observe that for the lengths 15, 63, and 255 there are eight distinct weight tistributions. We prove that this property holds for the codes B and B for all even m. Index Terms-BCH-codes, weight distribution of cosets, quadratic boolean functions, group algebra. p = max min { o(x + c)lc E E} L = F i X € L where w(x) is the weight of the codeword x. The code E is said to be perfect if p = e and quasiperfect if p = e + 1. The distance matrix of the code E is the 2' X (U + 1) matrix 9 (E) whose (x, j)-entry is 9 (x , j) = c a r d (y E x + Elw(y) = j } , x E E; .

Error-Correction Capability of Binary Linear Codes

IEEE Transactions on Information Theory, 2005

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense.

Weight distribution in some linear codes

2016

Paper describes the qualitative properties of linear codes in the small dimension and codimension, especially the same like oryginal Hamming code. The codes are constructed by algorithms based on „Monte Carlo” method. We considered four dimensions codes embedded in seven dimensional space over the field with twenty five elements. An analysis of the Hamming distance for such a codes is also represented in the paper.

Hamming Codes: Error Reducing Techniques

International Journal for Research in Applied Science and Engineering Technology (IJRASET) , 2021

Hamming codes for all intents and purposes are the first nontrivial family of error-correcting codes that can actually correct one error in a block of binary symbols, which literally is fairly significant. In this paper we definitely extend the notion of error correction to error-reduction and particularly present particularly several decoding methods with the particularly goal of improving the error-reducing capabilities of Hamming codes, which is quite significant. First, the error-reducing properties of Hamming codes with pretty standard decoding definitely are demonstrated and explored. We show a sort of lower bound on the definitely average number of errors present in a decoded message when two errors for the most part are introduced by the channel for for all intents and purposes general Hamming codes, which actually is quite significant. Other decoding algorithms are investigated experimentally, and it generally is definitely found that these algorithms for the most part improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of for all intents and purposes standard decoding.

Formalization of Error-Correcting Codes: From Hamming to Modern Coding Theory

Interactive Theorem Proving, 2015

By adding redundancy to transmitted data, error-correcting codes (ECCs) make it possible to communicate reliably over noisy channels. Minimizing redundancy and (de)coding time has driven much research, culminating with Low-Density Parity-Check (LDPC) codes. At first sight, ECCs may be considered as a trustful piece of computer systems because classical results are well-understood. But ECCs are also performance-critical so that new hardware calls for new implementations whose testing is always an issue. Moreover, research about ECCs is still flourishing with papers of ever-growing complexity. In order to provide means for implementers to perform verification and for researchers to firmly assess recent advances, we have been developing a formalization of ECCs using the SSReflect extension of the Coq proof-assistant. We report on the formalization of linear ECCs, duly illustrated with a theory about the celebrated Hamming codes and the verification of the sum-product algorithm for decoding LDPC codes.

A New Construction for Constant Weight Codes

A new construction for constant weight codes is presented. The codes are constructed from k-dimensional subspaces of the vector space F n q . These subspaces form a constant dimension code in a Grassmannian. Some of the constructed codes are optimal constant weight codes with parameters not known before. An efficient algorithm for error-correction is given for these codes. If the constant dimension code has an efficient encoding and decoding algorithm then also the constructed constant weight code has an efficient encoding and decoding algorithm.

Linear Codes with Non-Uniform Error Correction Capability

Designs, Codes and Cryptography, 1997

This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different codewords. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. This is accomplished by a new method called parity sectioned reduction. In this method, the parity check matrix of a uniform error correcting linear code is reduced by dropping some rows and columns and the error range inequalities are modified.