Related papers: Subfield subcodes of projective Reed-Muller codes
It is known that a linear code can be represented by a binomial ideal. In this paper, we give standard bases for the ideals in a localization of the multivariate polynomial ring in the case of linear codes over prime fields.
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for…
We study the classification of minimal codewords of projective Reed-Muller codes of order $2$. This problem is equivalent to identifying quadrics over finite fields whose set of rational points is maximal with respect to the inclusion. We…
Recently, Bhaintwal and Wasan studied the Generalized Reed-Muller codes over the prime power integer residue ring. In this paper, we give a generalization of these codes to Generalized Reed-Muller codes over Galois rings.
Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side,…
The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is…
In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the…
A framework of monomial codes is considered, which includes linear codes generated by the evaluation of certain monomials. Polar and Reed-Muller codes are the two best-known representatives of such codes and can be considered as two extreme…
Gr\"obner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field…
The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
Reed-Muller (RM) codes achieve the capacity of general binary-input memoryless symmetric channels and are conjectured to have a comparable performance to that of random codes in terms of scaling laws. However, such results are established…
We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult…
We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane…
We define and study a class of Reed-Muller type error-correcting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases.
Multivariate multiplicity codes have been recently explored because of their importance for list decoding and local decoding. Given a multivariate multiplicity code, in this paper, we compute its dimension using Gr\"obner basis tools, its…
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for…
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear…