Pairs in Nested Steiner Quadruple Systems

TL;DR

This paper introduces nested Steiner quadruple systems derived from classical SQS by partitioning blocks into pairs, exploring their properties, constructions, and uniformity conditions, motivated by distributed storage repair problems.

Contribution

It defines nested Steiner quadruple systems, investigates their multiplicity properties, and provides constructions and conditions for uniform nested pairs, extending classical SQS theory.

Findings

Maximum multiplicity of nested pairs with minimum multiplicity identified.

Conditions for all nested pairs to have the same multiplicity established.

Several explicit constructions of nested quadruple systems provided.

Abstract

Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order $v$ is partitioned into two pairs. Each pair in such a partition is called a nested design pair and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered as a new block design called a nested Steiner quadruple system. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the $\binom{v}{2}$ pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered and in particular classic constructions of SQS are examined.