Related papers: Group Divisible Codes and Their Application in the…
In this note, we give direct constructions of some group divisible designs (GDDs) with block size $4$ that have up to $50$ points.
Based on cyclic and consta-cyclic simplex codes, a new explicit construction of a family of two-weight codes is presented. These two-weight codes obtained are in the form of 2-generator quasi-cyclic, or quasi-twisted structure. Based on…
Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ans\"atze, unrestricted GB codes may have linear distance scaling. In…
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring…
We give direct constructions for 233 group divisible designs with block size five, mostly of type $g^u m^1$, $m > 0$.
An important family of codes for data storage systems, cryptography, consumer electronics, and network coding for error control in digital communications are the so-called cyclic codes. This kind of linear codes are also important due to…
Self-orthogonal codes are a subclass of linear codes that are contained within their dual codes. Since self-orthogonal codes are widely used in quantum codes, lattice theory and linear complementary dual (LCD) codes, they have received…
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…
We study codes with a single check element derived from group rings, namely, checkable codes. The notion of a code-checkable group ring is introduced. Necessary and sufficient conditions for a group ring to be code-checkable are given in…
In this paper, based on the theory of defining sets, two classes of at most six-weight linear codes over $\mathbb{F}_p$ are constructed. The weight distributions of the linear codes are determined by means of Gaussian period and Weil sums.…
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical…
Classes of self-dual codes and dual-containing codes are constructed. The codes are obtained within group rings and, using an isomorphism between group rings and matrices, equivalent codes are obtained in matrix form. Distances and other…
Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this paper is to give eight…
Codes considered as structures within unit schemes greatly extends the availability of linear block and convolutional codes and allows the construction of these codes to required length, rate, distance and type. Properties of a code emanate…
Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions…
Low check weight is practically crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we explore the theory of weight-constrained stabilizer…
New families of unit memory as well as multi-memory convolutional codes are constructed algebraically in this paper. These convolutional codes are derived from the class of group character codes. The proposed codes have basic generator…
We present constructions and results about GDDs with two groups and block size 6. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from…
We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…