On the next-to-minimal weight of affine cartesian codes

Designs, Codes and Cryptography, 2014

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.

Integrable Systems and Algebraic Geometry, 2020

In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed-Muller codes where, among many important results, they proved that the minimal weight codewords of these codes are obtained through the evaluation of certain polynomials which are a specific product of linear factors, which they describe. In the present paper we extend this result to a class of Reed-Muller type codes defined on a product of (possibly distinct) finite fields of the same characteristic. The paper also brings an expository section on the study of the structure of low weight codewords, not only for affine Reed-Muller type codes, but also for the projective ones.

Designs, Codes and Cryptography, 2014

We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation 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.

IEEE Transactions on Information Theory, 1998

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

Finite Fields and Their Applications, 2018

In this paper we present several values for the next-to-minimal weights of projective Reed-Muller codes. We work over F q with q ≥ 3 since in [3] we have determined the complete values for the next-to-minimal weights of binary projective Reed-Muller codes. As in [3] here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement.

Information and Control, 1976

IEEE Transactions on Information Theory, 1970

Journal of mechanics of continua and mathematical sciences, 2019

In this work we try to introduce the concept of codes of polynomial type and polynomial codes that are built over the ring A[X]/A[X]f(X).It should be noted that for particular cases of f we will find some classic codes for example cyclic codes, constacyclic codes, So the study of these codes is a generalization of linear codes.

Finite Fields and Their Applications, 2004

We generalize a recent idea for constructing codes over a finite field F q by evaluating a certain collection of polynomials over F q at elements of an extension field. We show that many codes with the best parameters presently known can be obtained by this construction. In particular, a new linear code, a ½40; 23; 10-code over F 5 is discovered. Moreover, several families of optimal and near-optimal codes can also be obtained by this method. We call a code near-optimal if its minimum distance is within 1 of the known upper bound.