A systematic convolutional encoder of rate $(n-1)/n$ and maximum memory $D$ generates a code of free distance at most ${\mathcal{ D}} = D+2$ and, at best, a column distance profile (CDP) of $[2,3,\ldots,{\mathcal{ D}}]$ . A code is memory maximum distance separable 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 , with a delay of exactly $j$ blocks counting from the first non-corrected erasure. This paper addresses the problem of finding the largest ${\mathcal{ D}}$ for which a systematic rate $(n-1)/n$ code over a given field exists, for given $n$ . In particular, constructions for rate $(p^{m}-1)/p^{m}$ and ${\mathcal{ D}}$ equal to 3 over $GF(p^{m})$ and rate $(2^{m-1}-1)/2^{m-1}$ and ${\mathcal{ D}}$ equal to 4 over $GF(2^{m})$ are presented, which provide optimum values of ${\mathcal{ D}}$ in their respective cases. A search algorithm is also developed, which produces new codes for ${\mathcal{ D}}$ for field sizes $2^{m} \leq 2^{14}$ . Using a complete search version of the algorithm, the maximum value of ${\mathcal{ D}}$ , and codes that achieve it, are determined for all code rates $\geq 1/2$ and every field size $GF(2^{m})$ for $m\leq 5$ (and for some rates for $m=6$ ).