Related papers: Harmonic higher and extended weight enumerators
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.…
In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the…
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…
In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting capacity. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the…
This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite Galois extension of a field $K$. This is a…
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…
We consider the question of determining the higher weights or the generalized Hamming weights of affine Grassmann codes and their duals. Several initial as well as terminal higher weights of affine Grassmann codes of an arbitrary level are…
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…
Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}.$ Necessary and sufficient…
In this paper, we prove the Gallai-Edmonds structure theorem for the most general matching polynomial. Our result implies the Parter-Wiener theorem and its recent generalization about the existence of principal submatrices of a Hermitian…
One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this…
The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient…
We describe the higher weights of the Grassmann codes $G(2,m)$ over finite fields ${\mathbb F}_q$ in terms of properties of Schubert unions, and in each case we determine the weight as the minimum of two explicit polynomial expressions in…
We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…
We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result…
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
An explicit formula for a weight enumerator of linear-congruence codes is provided. This extends the work of Bibak and Milenkovic [IEEE ISIT (2018) 431-435] addressing the binary case to the non-binary case. Furthermore, the extension…
In this paper, we introduce the concept of the complete cycle index and discuss a relation with the complete weight enumerator in coding theory. This work was motivated by Cameron's lecture note "Polynomial aspects of codes, matroids and…
In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators. The…
We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field…