Near-Optimal Covering Sequences

Abstract

An $(n,R)$-covering sequence over a finite alphabet $Σ_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $Σ_q^n$ is within Hamming distance $R$ of at least one window. We give a deterministic and explicit construction of such sequences whose length, for every fixed alphabet size $q$, every fixed radius $R$, and every sufficiently large $n$, attains the sphere-covering lower bound up to a constant factor depending only on $q$ and $R$. Thus, in the fixed-radius regime, the construction removes the logarithmic factor in the general probabilistic upper bounds of [Chung and Cooper, \emph{Random Structures \& Algorithms}, 2004] and [Vu, \emph{Advances in Applied Mathematics}, 2005]. It also complements the earlier explicit constructions of [Chee, Etzion, Ta, and Vu, \emph{Designs, Codes and Cryptography}, 2025], which include constant factor bounds for the special binary radius-one families \(n=2^a-1\) and \(n=2^a\), where \(a\ge1\).