相关论文: Density of normal binary covering codes
The length function $\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code of codimension $r$ and covering radius $R$. In this work we obtain new constructive upper bounds on $\ell_q(r,R)$ for all $R\ge4$, $r=tR$, $t\ge2$, and also…
Polycyclic codes offer a natural generalization of cyclic codes and provide a broader algebraic framework for constructing linear codes with good parameters. In this paper, we study binary polycyclic codes associated with powers of…
For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least…
The Universal Coding of Integers~(UCI) is suitable for discrete memoryless sources with unknown probability distributions and infinitely countable alphabet sizes. A UCI is a class of prefix codes for which the ratio of the average codeword…
We employ analytic number theoretic techniques, specifically character sums and Weil type estimates, to study the covering radius of the generalized Zetterberg codes over all finite fields. Although the even and odd field cases require…
The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It…
For any given alphabet of size $q$, a Homopolymer Free code (HF code) refers to an $(n, M, d)_q$ code of length $n$, size $M$ and minimum Hamming distance $d$, where all the codewords are homopolymer free sequences. For any given alphabet,…
A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2$, \ldots and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from…
A binary code is called a superimposed cover-free $(s,\ell)$-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of $\ell$ sets is covered by the union of $s$ others. A binary code is…
We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise…
Certain binary asymmetric channels, such as Z-channels in which one of the two crossover probabilities is zero, demand optimal ones densities different from 50%. Some broadcast channels, such as broadcast binary symmetric channels (BBSC)…
We investigate the asymptotic density of error-correcting codes with good distance properties and prescribed linearity degree, including sublinear and nonlinear codes. We focus on the general setting of finite translation-invariant metric…
We consider the isoperimetric problem in $\mathbb R^n$ with density for the planar case $n=2$. We show that, if the density is ${\rm C}^{0,\alpha}$, then the boundary of any isoperimetric is of class ${\rm C}^{1,\frac \alpha{3-2\alpha}}$.…
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the…
We show that any q-ary code with sufficiently good distance can be randomly punctured to obtain, with high probability, a code that is list decodable up to radius $1 - 1/q - \epsilon$ with near-optimal rate and list sizes. Our results imply…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of $O(1/\gamma)$) and lower bound (of $\Omega_p(\log (1/\gamma))$) for the list-size needed to decode up to radius $p$…
Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques…
We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold…
In a {\em locally recoverable} or {\em repairable} code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed…