A family of diameter perfect constant-weight codes from Steiner systems

TL;DR

This paper constructs diameter perfect constant-weight codes from Steiner systems, achieving optimal bounds and analyzing the minimal alphabet size needed for their existence.

Contribution

It introduces a new family of diameter perfect codes derived from Steiner systems and investigates the minimal alphabet size for their construction.

Findings

Existence of diameter perfect codes from Steiner systems for specific parameters.

Construction of codes that attain the code--anticode bound.

Determination of minimal alphabet size for small parameter values.

Abstract

If $S$ is a transitive metric space, then $|C|\cdot|A| \le |S|$ for any distance-$d$ code $C$ and a set $A$, ``anticode'', of diameter less than $d$. For every Steiner S$(t,k,n)$ system $S$, we show the existence of a $q$-ary constant-weight code $C$ of length~$n$, weight~$k$ (or $n-k$), and distance $d=2k-t+1$ (respectively, $d=n-t+1$) and an anticode $A$ of diameter $d-1$ such that the pair $(C,A)$ attains the code--anticode bound and the supports of the codewords of $C$ are the blocks of $S$ (respectively, the complements of the blocks of $S$). We study the problem of estimating the minimum value of $q$ for which such a code exists, and find that minimum for small values of $t$. Keywords: diameter perfect codes, anticodes, constant-weight codes, code--anticode bound, Steiner systems.