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

## TL;DR

This paper investigates the maximum free distance and column distance profile of rate (n-1)/n systematic convolutional codes over GF(2^m), providing constructions, algorithms, and optimal codes for various parameters.

## Contribution

It introduces new constructions and a search algorithm to find optimal systematic convolutional codes with maximum free distance over GF(2^m).

## Key findings

- Optimal codes with D=3 for rate (2^m-1)/2^m.
- Optimal codes with D=4 for rate (2^{m-1}-1)/2^{m-1}.
- Complete classification of maximum D for rates ≥ 1/2 and m ≤ 5.

## Abstract

A systematic convolutional encoder of rate $(n-1)/n$ and maximum degree $D$ generates a code of free distance at most ${\cal D} = D+2$ and, at best, a column distance profile (CDP) of $[2,3,\ldots,{\cal D}]$. A code is \emph{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 $j<{\cal D}$, with a delay of at most $j$ blocks counting from the first erasure. This paper addresses the problem of finding the largest ${\cal D}$ for which a systematic rate $(n-1)/n$ code over $GF(2^m)$ exists, for given $n$ and $m$. In particular, constructions for rates $(2^m-1)/2^m$ and $(2^{m-1}-1)/2^{m-1}$ are presented which provide optimum values of ${\cal D}$ equal to 3 and 4, respectively. A search algorithm is also developed, which produces new codes for ${\cal D}$ for field sizes $2^m \leq 2^{14}$. Using a complete search version of the algorithm, the maximum value of ${\cal D}$, 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 $m=6$).