Related papers: On the maximum size of variable-length non-overlap…

We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $q^r$-divisible codes over $\mathbb{F}_q$. We also give a few…

\textit{Matching families} are one of the major ingredients in the construction of {\em locally decodable codes} (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The…

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding…

We improve on the lower bound of the maximum number of planes in $\operatorname{PG}(8,q)\cong\F_q^{9}$ pairwise intersecting in at most a point. In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+…

A code is called a $q$-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index $i$ and a received word $w$ close to an encoding of a message $x$, outputs $x_i$ by querying only at most $q$…

A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It…

A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate…

A quasi-Gray code of dimension $n$ and length $\ell$ over an alphabet $\Sigma$ is a sequence of distinct words $w_1,w_2,\dots,w_\ell$ from $\Sigma^n$ such that any two consecutive words differ in at most $c$ coordinates, for some fixed…

Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected…

Let $R=\mathbb{Z}_{4}[u]/\langle u^k\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of…

One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $\mathcal{P}_q(n)$ for a given minimum distance. The…

A $q$-ary $t$-covering array is an $m \times n$ matrix with entries from $\{0, 1, ..., q-1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for…

Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$,…

Variable-length compression without prefix-free constraints and with side-information available at both encoder and decoder is considered. Instead of requiring the code to be error-free, we allow for it to have a non-vanishing error…

Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The error detection…

A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-\epsilon)n$ has the property that every Hamming ball of radius $pn$ has…

A $(p,q)$-ary chain is a special type of chain partition of integers with parts of the form $p^aq^b$ for some fixed integers $p$ and $q$. In this note, we are interested in the maximal weight of such partitions when their parts are distinct…

A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been…

We investigate inference of variable-length codes in other domains of computer science, such as noisy information transmission or information retrieval-storage: in such topics, traditionally mostly constant-length codewords act. The study…