Related papers: The second weight of generalized Reed-Muller codes…
In this paper we give the second weight codewords of the generalized Reed-Muller code of order r and length $q^m$.
In this paper, we study the third weight of generalized Reed-Muller codes. We prove under some restrictive condition that the third weight of generalized Reed-Muller codes depends on the third weight of generalized Reed-Muller codes of…
Determining the weight distributions of the projective Reed-Muller codes is a very hard problem and has been studied extensively in the literature. In this article, we provide an alternative proof of the second weight of the projective…
The weight distribution of second order $q$-ary Reed-Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for general…
In [2] we show how to construct information sets for Reed-Muller codes only in terms of their basic parameters. In this work we deal with the corresponding problem for q-ary Generalized Reed-Muller codes of first and second order. We see…
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a…
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.…
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…
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…
The weight hierarchy of generalized Reed-Muller codes over arbitrary finite fields was determined by Heijnen and Pellikaan. In this paper we produce curves over finite fields with many points which are closely related to this weight…
Linearized Reed-Solomon codes are defined. Higher weight distribution of those codes are determined.
In this paper, we first introduce the notion of generalized pair weights of an $[n, k]$-linear code over the finite field $\mathbb{F}_q$ and the notion of pair $r$-equiweight codes, where $1\le r\le k-1$. Some basic properties of…
We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$…
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight…
We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…
We consider the weight distributions of the cosets of weight 2 of the generalized $[q+1,q+2-d,d]_q$ doubly extended Reed-Solomon codes (GDRS) of minimum distance $d\ge5$, over the finite field $\mathbb{F}_q$ with $q$ elements. For a GDRS…
For any q which is a power of 2 we describe a finite subgroup of the group of invertible complex q by q matrices under which the complete weight enumerators of generalized doubly-even self-dual codes over the field with q elements are…
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…
We use a simple construction called `recursive subproducts' (that is known to yield good codes of lengths $n^m$, $n \geq 3$) to identify a family of codes sandwiched between first-order and second-order Reed-Muller (RM) codes. These codes…
Over a finite field $\F_q$ the $(n,d,q)$-Reed-Muller code is the code given by evaluations of $n$-variate polynomials of total degree at most $d$ on all points (of $\F_q^n$). The task of testing if a function $f:\F_q^n \to \F_q$ is close to…