Related papers: On Hull-Variation Problem of Equivalent Linear Cod…

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean…

This paper explores extremal self-dual double circulant (DC) codes and linear complementary dual (LCD) codes of arbitrary length over the Galois field $\mathbb F_2$. We establish the sufficient and necessary conditions for DC codes and…

Ternary channels can be used to model the behavior of some memory devices, where information is stored in three different levels. In this paper, error correcting coding for a ternary channel where some of the error transitions are not…

Characterizing the duals of linear codes with rich algebraic structures received great interest in recent decades. The beginning was by representing cyclic codes over finite fields as ideals in the polynomial ring. Subsequently, studying…

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal…

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,…

Projective Reed-Muller codes(PRM codes) are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. In this paper, we completely determine the minimal distance of the hull of any Projective…

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in…

Based on the theoretical neuroscience, G. Cotardo and A. Ravagnavi in \cite{CR} introduced a kind of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a "matched metric" $\delta_{r}$ called asymmetric…

The Schur square of linear codes over a finite field has emerged as a fundamental operation in both classical and quantum coding theory. In this paper, we investigate the Schur square problem of Hyperderivative Reed-Solomon (HRS) codes. By…

Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of…

Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is…

We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to…

We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$…

An equi-differece conflict-avoiding code $(CAC^{e})\ \mathcal{C}$ of length $n$ and weight $\omega$ is a collection of $\omega$-subsets (called codewords) which has the form $\{0,i,2i,\cdots,(\omega-1)i\}$ of $\mathbb{Z}_{n}$ such that…

Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a…

We establish the existence of optimal maximal entanglement entanglement-assisted quantum $[[n,k,d;n-k]]_2$ codes for $(n,k,d)=(14,6,7)$, $(15,7,7)$, $(17,6,9)$, $(17,7,8)$, $(19,7,9)$ and $(20,7,10)$. These codes are obtained from…

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let $S_n$ denote the set of permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their Ulam distance…

Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes over $\F_q$ which are self-orthogonal with respect to…