Problem
Research questions and friction points this paper is trying to address.
limited magnitude error-correcting codes
splitter sets
group splittings
quasi-perfect codes
burst-correcting codes
New Results on Limited Magnitude Error Correcting Codes
🤖 AI Summary
This study addresses the problem of efficient error correction for bounded-magnitude errors—such as charge leakage—in flash memory by investigating the existence and construction of quasi-perfect and perfect splitting sets. By integrating group splitting theory, algebraic combinatorics, number theory, and Cayley graph analysis, the work establishes an algebraic connection between splitting sets and cyclic burst-error-correcting codes. The main contributions include the first complete classification of quasi-perfect $B[0,3](n)$ splitting sets in both singular and non-singular cases, a general construction for an infinite family of $(k_2,k_1)$-bounded cyclic $b$-burst error-correcting codes, sufficient conditions for their existence, new non-existence results, and an improved lower bound on the size of maximal $B[0,3](q)$ sets—collectively enabling the design of highly efficient burst-error-correcting code families for arbitrary burst lengths.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
limited magnitude error-correcting codes
splitter sets
quasi-perfect codes
Cayley graphs
burst-correcting codes
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