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

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. They 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. ∗The authors are with Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands; e-mail:{henk.d.l.hollmann,ludo.tolhuizen}@philips.com 1