Related papers: A note on reducing the computation time for minimu…
The AWGNC, BSC, and max-fractional pseudocodeword redundancies of a binary linear code are defined to be the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming…
A self-dual binary linear code is called Type I code if it has singly-even codewords, i.e.~it has codewords with weight divisible by $2.$ The purpose of this paper is to investigate interesting properties of Type I codes of different…
We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an…
A Gray code for a combinatorial class is a method for listing the objects in the class so that successive objects differ in some prespecified, small way, typically expressed as a bounded Hamming distance. In a previous work, the authors 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…
For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight…
Bounds on linear codes play a central role in coding theory, as they capture the fundamental trade-off between error-correction capability (minimum distance) and information rate (dimension relative to length). Classical results…
We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed…
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 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.
Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating…
Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realized by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free…
In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved…
We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is…
A constant weight binary code consists of $n$-bit binary codewords, each with exactly $w$ bits equal to 1, such that any two codewords are at least Hamming distance $d$ apart. $A(n,d,w)$ is the maximum size of a constant weight binary code…
Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in…
Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power…
In this paper we study codes for correcting deletable errors in binary words, where each bit is either retained, substituted, erased or deleted and the total number of errors is much smaller compared to the length of the codeword. We…
In this article, we construct linear codes over the commutative non-unital ring $I$ of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are…
Determining the largest size, or equivalently finding the lowest redundancy, of q-ary codes for given length and minimum distance is one of the central and fundamental problems in coding theory. Inspired by the construction of…