AI summaryⓘ
The authors study special types of error-correcting codes that handle small errors in data storage, focusing on 'splitter sets' and how they relate to mathematical group structures. They discover new cases where certain optimal splitter sets cannot exist and fully classify some specific types of these sets. By examining mathematical graphs, they improve estimates on how large these sets can be when based on prime numbers. They also develop a general method to build codes that correct bursts of errors and prove that infinite families of such codes exist under certain conditions. Their work uses a mix of algebra, combinatorics, and number theory and is relevant for improving data reliability in devices like flash memory.
limited magnitude error-correcting codessplitter setsgroup splittingsquasi-perfect codesCayley graphsprime numbersburst-correcting codescyclic codesalgebraic coding theorycombinatorics
Authors
Zhiyu Yuan, Tingting Chen, Rongquan Feng, Gennian Ge
Abstract
This paper investigates the existence, construction and classification of limited magnitude error-correcting codes, with a focus on splitter sets and their connections to group splittings. We establish new nonexistence results for quasi-perfect splitter sets and provide a complete classification of quasi-perfect $B[0,3](n)$ splitter sets in both singular and nonsingular cases. Furthermore, we derive improved lower bounds for the size of maximal $B[0,3](q)$ sets by investigating Cayley graphs, where $q$ is a prime. We also provide existence criteria for perfect $B[0,6](q)$ splitter sets and quasi-perfect $B[-4,4](2p)$ sets for prime $p$. For perfect burst-correcting codes, we develop a general construction framework, and prove the existence of infinite families of $(k_2,k_1)$-limited-magnitude cyclic $b$-burst-correcting codes for $k_1+k_2\le 4$ and arbitrary burst length $b$. We further provide sufficient existence conditions for general parameters $k_1$ and $k_2$. Our results combine algebraic, combinatorial, and number-theoretic methods to advance the understanding of codes tailored for flash memory and related storage systems.