Some New Results on Splitter Sets

TL;DR

This paper investigates the existence and construction of splitter sets with applications in flash memories, providing necessary and sufficient conditions, new constructions, and exploring their relation to Cayley graphs.

Contribution

It establishes conditions for perfect splitter sets, introduces new constructions for both perfect and quasi-perfect sets, and links splitter sets to Cayley graphs with a lower bound on their size.

Findings

Necessary and sufficient conditions for nonsingular perfect splitter sets.

New constructions of quasi-perfect splitter sets.

A lower bound on the maximum size of nonsingular splitter sets.

Abstract

Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of nonsingular perfect splitter sets, $B[-k_1,k_2](p)$ sets, where $0\le k_{1}\leq k_{2}=4$. Meanwhile, constructions of nonsingular perfect splitter sets are given. When perfect splitter sets do not exist, we present four new constructions of quasi-perfect splitter sets. Finally, we give a connection between nonsingular splitter sets and Cayley graphs, and as a byproduct, a general lower bound on the maximum size of nonsingular splitter sets is given.