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Pairs in Nested Steiner Quadruple Systems

Yeow Meng Chee, Son Hoang Dau, Tuvi Etzion, Han Mao Kiah, Wenqin Zhang

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.

Pairs in Nested Steiner Quadruple Systems

Abstract

Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order 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 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.

Paper Structure

This paper contains 5 sections, 27 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Bounds Concerning the Nested Design Pairs
  3. Recursive Constructions
  4. Blocks Partition with Uniform Multiplicity
  5. Conclusions and Problems for Future Research

Key Result

Lemma 2.1

Any pair $\{x,y\}$ in a block of an SQS$(v)$, is contained in $\frac{v-2}{2}$ blocks, i.e., the multiplicity of an ND-pair in the nested SQS is at most $\frac{v-2}{2}$.

Theorems & Definitions (55)

  • Example 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • ...and 45 more