Related papers: Computing Extensions of Linear Codes
The square $C^{*2}$ of a linear error correcting code $C$ is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in $C$. Squares of codes have gained attention for several applications…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial…
In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In…
Echelon-Ferrers is an important method to improve lower bounds for constant-dimension codes, which can be applied on various parameters. Fagang Li [12] combined the linkage construction and echelon-Ferrers to obtain some new lower bounds of…
An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix…
In this paper, we present improved decoding algorithms for expander-based Tanner codes. We begin by developing a randomized linear-time decoding algorithm that, under the condition that $ \delta d_0 > 2 $, corrects up to $ \alpha n $ errors…
The groupcast index coding problem is the most general version of the classical index coding problem, where any receiver can demand messages that are also demanded by other receivers. Any groupcast index coding problem is described by its…
In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF(q). The comparison to the lower bound…
The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with…
The minimum number of elements needed for a poset to have exactly n linear extensions is at most 2sqrt{n}. In a special case, the bound can be improved to sqrt{n}.
The minimum distance of a linear code is a key concept in information theory. Therefore, the time required by its computation is very important to many problems in this area. In this paper, we introduce a family of implementations of the…
Linear complementary dual (LCD) maximum distance separable (MDS) codes are constructed to given specifications. For given $n$ and $r<n$, with $n$ or $r$ (or both) odd, MDS LCD $(n,r)$ codes are constructed over finite fields whose…
Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly…
The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for…
A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes,…
Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional…
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights…
We explain how to optimize finite-length LDPC codes for transmission over the binary erasure channel. Our approach relies on an analytic approximation of the erasure probability. This is in turn based on a finite-length scaling result to…
We propose an algorithm using the Gaussian elimination method to find the minimal Hamming distance and decode received messages of linear codes. This algorithm is easy to implement as it requires no Gr\"obner bases to compute solutions for…