On the Existence and Nonexistence of Splitter Sets

TL;DR

This paper investigates the existence of perfect and quasi-perfect splitter sets in finite abelian groups, providing new conditions and nonexistence results relevant to coding theory applications.

Contribution

It introduces a general condition for splitter set existence using cyclotomic polynomials and establishes new relations and nonexistence results for specific splitter sets.

Findings

Derived a general existence condition for splitter sets using cyclotomic polynomials.

Established a relation between different types of splitter sets, specifically $B[-k, k](q)$ and $B[-(k-1), k+1](q)$.

Presented nonexistence results for certain quasi-perfect splitter sets.

Abstract

In this paper, the existence of perfect and quasi-perfect splitter sets in finite abelian groups is studied, motivated by their application in coding theory for flash memory storage. For perfect splitter sets we view them as splittings of $\mathbb{Z}_n$, and using cyclotomic polynomials we derive a general condition for the existence of such splittings under certain circumstances. We further establish a relation between $B[-k, k](q)$ and $B[-(k-1), k+1](q)$ splitter sets, and give a necessary and sufficient condition for the existence of perfect $B[-1, 5](q)$ splitter sets. Finally, two nonexistence results for quasi-perfect splitter sets are presented.