Related papers: An improved method for constructing linear codes w…
The minimum distance of a code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is a crucial process to many problems in this area. In this paper, we present…
Reading channels where $b$-tuples of adjacent symbols are read at every step have e.g.\ applications in storage. Corresponding bounds and constructions of codes for the $b$-symbol metric, especially the pair-symbol metric where $b=2$, were…
We present simple constructions of optimal erasure-correcting LRC codes by exhibiting their parity-check matrices. When the number of local parities in a parity group plus the number of global parities is smaller than the size of the parity…
In an $[n,k,d]$ linear code, a code symbol is said to have locality $r$ if it can be repaired by accessing at most $r$ other code symbols. For an $(n,k,r)$ \emph{locally repairable code} (LRC), the minimum distance satisfies the well-known…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of…
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…
This paper proposes new propagation rules on quantum codes in the entanglement-assisted and in quantum subsystem scenarios. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the…
The (n, k, r)-Locally recoverable codes (LRC) studied in this work are (n, k) linear codes for which the value of each coordinate can be recovered by a linear combination of at most r other coordinates. In this paper, we are interested to…
Products of MDS codes are of major practical importance; for a recent example, they are used in Data Availability Sampling (DAS) in blockchain networks such as Celestia and as part of the Ethereum roadmap. This motivates us to consider…
Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of…
In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that…
We introduce quaternary modified four $\mu$-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four $\mu$-circulant Hermitian self-dual codes. We also construct quaternary modified…
We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum…
Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains…
In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes -- a class of linear codes -- have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic…
The construction of deletion codes for the Levenshtein metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of translates of…
We construct MDS Euclidean and Hermitian self-dual codes over large finite fields of odd and even characteristics. Our codes arise from cyclic and negacyclic duadic codes.
In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of \cite{ballentine2019codes,sasidharan2015codes} with an algebraic number theoretical approach that…
Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…