Optimal codes for correcting a single (wrap-around) burst of errors

TL;DR

This paper proves the existence of optimal codes capable of correcting single wrap-around burst errors for all parameters, providing recursive and direct constructions to achieve this.

Contribution

It introduces two recursive and one direct construction method for optimal codes correcting single wrap-around burst errors across all parameters.

Findings

Existence of such codes for all parameters is confirmed.

Provides explicit recursive constructions for these codes.

Offers a direct construction method for the codes.

Abstract

In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs [n,k] codes that can correct any burst of erasures (including wrap-around bursts) of length n-k. The raised the question if such [n,k] codes exist for all integers k and n with 1<= k <= n and all fields (in particular, for the binary field). In this note, we answer this question affirmatively by giving two recursive constructions and a direct one.