Rate $(n-1)/n$ Systematic MDS Convolutional Codes over $GF(2^m)$

A systematic convolutional encoder of rate (n− 1)/n and maximum degree D generates a code of free distance at most D = D+ 2 and, at best, a column distance profile (CDP) of [2,3, . . . ,D ]. A code is Maximum Distance Separable (MDS) if it possesses this CDP. Applied on a communication channel over which packets are transmitted sequentially and which loses (erases) packets randomly, such a code allows the recovery from any pattern of j erasures in the first j n-packet blocks for j < D , with a delay of at most j blocks counting from the first erasure. This paper addresses the problem of finding the largest D for which a systematic rate (n− 1)/n code over GF(2m) exists, for given n and m. In particular, constructions for rates (2m− 1)/2m and (2m−1− 1)/2m−1 are presented which provide optimum values of D equal to 3 and 4, respectively. A search algorithm is also developed, which produces new codes for D for field sizes 2m ≤ 214. Using a complete search version of the algorithm, the maximum value of D , and codes that achieve it, are determined for all code rates ≥ 1/2 and every field size GF(2m) for m≤ 5 (and for some rates for m = 6).