Tilings of $\mathcal{H}_{q}(n,w)$ with optimal $(n,d,w)_{q}$-codes
Yuli Tan, Junling Zhou
TL;DR
The paper studies tilings of the weight-$w$ Hamming space $oldsymbol{H}_{q}(n,w)$ by mutually disjoint optimal $(n,d,w)_{q}$-codes, formalized as $ ext{TOC}_{q}(n,d,w)$. It builds a bridge between coding theory and combinatorial designs by tying odd distances to large sets of generalized Steiner systems and even distances to generalized maximum $H$-packings, and develops constructions via $t$-resolvable Steiner systems and almost-regular edge-colorings. The authors provide complete existence results for $d=2$ and $d=2w$, plus a rich collection of weight-$3$ tilings: binary cases are fully resolved and many infinite families for $q eq 2$ are obtained for $d=3,4,5$, with general methods applicable to broader parameters. These tilings advance understanding of how to partition $oldsymbol{H}_{q}(n,w)$ into optimal constant-weight codes, offering both theoretical insights and potential applications in powerline communications, DNA computing, and storage systems.
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 $d=3,4,5$.