Related papers: Constructions and Bounds for Mixed-Dimension Subsp…
An erasure channel with a fixed alphabet size $q$, where $q \gg 1$, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming…
This paper introduces the \emph{$d$-distance matching problem}, in which we are given a bipartite graph $G=(S,T;E)$ with $S=\{s_1,\dots,s_n\}$, a weight function on the edges and an integer $d\in\mathbb Z_+$. The goal is to find a maximum…
This paper concerns non-overlapping codes, block codes motivated by synchronisation and DNA-based storage applications. Most existing constructions of these codes do not account for the restrictions posed by the physical properties of…
We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular,…
Let $\chi'_d(n,q)$ (resp. $\chi_d(n,q)$) denote the minimum number of colors necessary to color a $q$-ary $n$-cube so that no two vertices that are at a distance at most $d$ (resp. exactly $d$) get the same color. These two problems were…
The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A…
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…
New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for q <= 9109. From these new bounds it follows that for q <= 2621 and q = 2659,2663,2683,2693,2753,2801, the relation…
Consider the geometric range space $(X, \mathcal{H}_d)$ where $X \subset \mathbb{R}^d$ and $\mathcal{H}_d$ is the set of ranges defined by $d$-dimensional halfspaces. In this setting we consider that $X$ is the disjoint union of a red and…
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
A locally repairable code is called Singleton-optimal if it achieves the Singleton-type bound. Such codes are of great theoretic interest in the study of locally repairable codes. In the recent years there has been a great amount of work on…
Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and…
The problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught…
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…
Conflict-avoiding codes (CACs) have been used in multiple-access collision channel without feedback. The size of a CAC is the number of potential users that can be supported in the system. A code with maximum size is called optimal. The use…
We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory…
Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\} $ and $\{B_1,B_2\} $ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap…
In this paper, we consider the problem of constructing optimal average-length binary codes under the constraint that each codeword must contain at most $D$ ones, where $D$ is a given input parameter. We provide an $O(n^2D)$-time complexity…
There is a known best possible upper bound on the probability of undetected error for linear codes. The $[n,k;q]$ codes with probability of undetected error meeting the bound have support of size $k$ only. In this note, linear codes of full…
In [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces…