Constructions of Covering Sequences and Arrays

TL;DR

This paper introduces new constructions for covering sequences and arrays, improving bounds for small parameters and generalizing the concept to sets of sequences and arrays, showing asymptotic optimality.

Contribution

It presents novel construction methods for covering sequences and arrays, extends the definitions to sets and arrays, and proves asymptotic bounds close to the sphere-covering limit.

Findings

Improved upper bounds for covering sequences with small parameters

Generalized definitions to sets of sequences and arrays

Asymptotic existence of near-optimal covering sequences

Abstract

An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. Using several construction methods improvements of the upper bounds on the length of such sequences for $n \leq 20$ and $1 \leq R \leq 3$, are obtained. The definition is generalized in two directions. An $(n,m,R)$-covering sequence code is a set of cyclic sequences of length $m$ whose consecutive $n$-tuples form a code of length~$n$ and covering radius $R$. The definition is also generalized to arrays in which the $m \times n$ sub-matrices form a covering code with covering radius $R$. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.