Related papers: Latroids and code invariants
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of…
We provide a combinatorial construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an arbitrary number of distinct…
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…
The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and…
We introduce the notion of sum-matroids and show its association with sum-rank metric codes. As a consequence, some results for sum-rank metric codes by Mart\'inez-Pe\~nas are generalized for sum-matroids. The sum-matroids generalize the…
The index coding problem has been generalized recently to accommodate receivers which demand functions of messages and which possess functions of messages. The connections between index coding and matroid theory have been well studied in…
In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code…
The weight distribution and weight hierarchy of linear codes are two important research topics in coding theory. In this paper, by choosing proper defining sets from inhomogeneous quadratic functions over $\mathbb{F}_{q}^{2},$ we construct…
Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes…
In the past few years, linear codes with few weights and their weight analysis have been widely studied. In this paper, we further investigate a class of two-weight or three-weight linear codes from defining sets and determine their weight…
Firstly, we give a formula on the generalized Hamming weight of linear codes constructed generically by defining sets. Secondly, by choosing properly the defining set we obtain a class of cyclotomic linear codes and then present two…
The Tutte polynomial of a connected graph was originally defined by Tutte as a sum over all spanning trees of monomials depending on a fixed linear order on the set of edges. Tuttle proved that while these monomials do depend on the linear…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…
We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over…
We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial $h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a prime number. These…
In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with…
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages…
Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of linear codes has many applications such as on the type II wire-tap channel, dealing with…
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are…