Steiner Systems over Mixed Alphabet and Related Designs

TL;DR

This paper introduces mixed Steiner systems over varying alphabets, explores their construction from existing combinatorial designs, and establishes necessary conditions and non-existence results for large sets of such systems.

Contribution

It defines mixed Steiner systems over mixed alphabets, links them to various combinatorial structures, and provides foundational existence and non-existence results.

Findings

Constructed from perfect mixed codes and resolvable designs

Established necessary conditions for existence

Proved non-existence of large sets of these Steiner systems

Abstract

A mixed Steiner system MS$(t,k,Q)$ is a set (code) $C$ of words of weight $k$ over an alphabet $Q$, where not all coordinates of a word have the same alphabet size, each word of weight $t$, over $Q$, has distance $k-t$ from exactly one codeword of $C$, and the minimum distance of the code $2(k-t)+1$. Mixed Steiner systems are constructed from perfect mixed codes, resolvable designs, large set, orthogonal arrays, and a new type of pairs-triples design. Necessary conditions for the existence of mixed Steiner systems are presented and it is proved that there are no large sets of these Steiner systems.