Tilings of $\mathcal{H}_{q}(n,w)$ with optimal $(n,d,w)_{q}$-codes

TL;DR

This paper studies how to partition the set of all weight-w words over a finite alphabet into optimal constant-weight codes with specific distances, providing complete solutions for certain parameters and constructing many new code families.

Contribution

It introduces new methods for constructing tilings of the Hamming space with optimal codes, resolving existence for key cases and expanding known code families.

Findings

Complete existence results for d=2 and d=2w cases.

Resolution of the existence problem for weight three codes.

Construction of infinite families of codes for q≥3 and distances 3,4,5.

Abstract

The metric space $\mathcal{H}_{q}(n,w)$ is the set of all words of length $n$ with weight $w$ over the alphabet $\mathbb{Z}_{q}$, under the Hamming distance metric. A $q$-ary constant-weight code, as a nonempty subset of $\mathcal{H}_{q}(n,w)$, has always been a fundamental topic in coding theory. This paper investigates the tiling problem of $\mathcal{H}_{q}(n,w)$ with optimal $(n,d,w)_{q}$-codes, simply denoted by $\mathrm{TOC}_{q}(n,d,w)$, meaning a partition of $\mathcal{H}_{q}(n,w)$ into mutually disjoint optimal $q$-ary constant-weight codes with distance $d$. When the distance $d$ is odd, we investigate large sets of generalized Steiner systems. When $d$ is even, we define large sets of generalized maximum H-packings. We present several general construction approaches for generating $\mathrm{TOC}_{q}(n,d,w)$s via $t$-resolvable Steiner systems and almost-regular edge-colorings of complete hypergraphs. For the cases $d=2$ and $d=2w$, we completely resolve the existence problem of $\mathrm{TOC}_{q}(n,d,w)$s for all parameters $q,n$ and $w$. Particularly, we pay attention to tilings for weight three. For binary case and weight three, the existence problem of $\mathrm{TOC}_{2}(n,d,3)$s is totally resolved. For specific alphabet size $q\ge 3$, we obtain many infinite families of $\mathrm{TOC}_{q}(n,d,3)$s for distances $d=3,4,5$.