Perfect Codes Correcting a Single Burst of Limited-Magnitude Errors

TL;DR

This paper introduces new perfect codes capable of correcting single bursts of limited-magnitude errors, with constructions applicable to DNA storage, flash memory, and magnetic recording, expanding the theoretical understanding of error correction in these channels.

Contribution

The paper presents two classes of perfect burst-correcting codes for limited-magnitude errors and a generic construction based on finite field primitive elements, proving their existence in many parameter regimes.

Findings

Constructed two classes of perfect codes for single burst correction.

Provided a generic finite field-based construction for such codes.

Proved the existence of infinitely many perfect codes in various parameters.

Abstract

Motivated by applications to DNA-storage, flash memory, and magnetic recording, we study perfect burst-correcting codes for the limited-magnitude error channel. These codes are lattices that tile the integer grid with the appropriate error ball. We construct two classes of such perfect codes correcting a single burst of length $2$ for $(1,0)$-limited-magnitude errors, both for cyclic and non-cyclic bursts. We also present a generic construction that requires a primitive element in a finite field with specific properties. We then show that in various parameter regimes such primitive elements exist, and hence, infinitely many perfect burst-correcting codes exist.