Error-Correcting Codes With Large Field Size Under Non-Binary Segmented Burst Deletion/Insertion Channels and Unknown Codeword Boundaries

In this paper, we construct non-binary codes of length N which correct errors under a non-binary segment-N maxburst-D deletion and maxburst-S insertion (NB-SBDI( $N,D,S$ )) channel without knowing the codeword boundaries. In this NB-SBDI( $N,D,S$ ) channel, at most a single non-binary burst (a block of consecutive bits/symbols) of deletions or insertions of length up to D or S, respectively, exists in a block of N consecutive non-binary symbols. One code named as BM-DB-MDS consists of a maximum distance separable (MDS) code, a block of periodic de Bruijn (DB) symbols, and a block of proposed periodic binary marker (BM) patterns with a period of $S+D+2$ . The other code called BM-MDS code consists of a BM code and an MDS code. We show that the rates of BM-DB-MDS and BM-MDS codes achieve $\frac {\lambda }{\lambda +1}\left ({{1-\frac {1}{2t}}}\right)$ , $\lambda \in \mathbb {N}^{+}$ and $1-\frac {1}{t}$ , respectively, when $N\rightarrow +\infty $ , where $\lambda $ represent the MDS code shortening factor, and $\frac {1}{t}$ is the rough proportion of the maximum length of burst deletions or insertions allowed in a code.