Correcting Non-Binary Burst Deletions/Insertions with De Bruijn Symbol-Maximum Distance Separable Codes

Abstract

We propose a deterministic decoding scheme to correct a block of deletions/insertions of length up to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> - <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> = ⌈0.5π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> ⌉ - 1 with a high probability for the de Bruijn symbol-maximum distance separable (DB-MDS( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N,K</i> , π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> )) codes. The DB-MDS( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N,K</i> , π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> ) codes are constructed by combining the MDS( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N,K</i> ) (maximum distance separable) codes and the de Bruijn (DB) symbol sequence with both the period and field size being π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> . The type, length, and positions of burst errors are first guessed and checked and the corrupted DB-MDS code symbol sequence is synchronized, then the errors introduced in the synchronization stage are corrected by the MDS code component of the DB-MDS code. We provide the closed form of the decoding failure for the proposed synchronization algorithm, which decreases as the code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> increases.