On the Existence and Nonexistence of Splitter Sets

2025-07-09

AI Summary

This study addresses the existence of perfect and quasi-perfect splitting sets in finite abelian groups—a problem with significant applications in flash memory coding. Using cyclotomic polynomial theory, we conduct a structural analysis of splittings over ℤₙ, integrating tools from algebraic number theory and combinatorial design. Our contributions are threefold: (i) we establish a novel structural correspondence between B[-k,k](q) and B[-(k−1),k+1](q) splitting sets; (ii) we derive the first necessary and sufficient condition for the existence of perfect B[-1,5](q) splitting sets; and (iii) we rigorously prove the nonexistence of two canonical classes of quasi-perfect splitting sets in all prime-power-order groups. Collectively, these results provide a general existence criterion for perfect splitting sets under specific parameters and fill a critical gap in the theoretical understanding of quasi-perfect splittings—thereby advancing and systematizing the foundational framework of splitting set theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Existence of perfect splitter sets in abelian groups
• Relation between different types of splitter sets
• Nonexistence conditions for quasi-perfect splitter sets

Innovation

• Uses cyclotomic polynomials for splitter sets
• Links B[-k, k] and B[-(k-1), k+1] splitter sets
• Provides existence conditions for perfect splittings