Optimal Coding for the Erasure Channel with Arbitrary Alphabet Size

TL;DR

This paper demonstrates that MDS codes are optimal for erasure channels with large alphabet sizes, achieving minimal error probability and matching the error exponents of random and linear codes at high rates.

Contribution

It proves the optimality of MDS codes over any erasure channel and compares their error exponents with those of random and linear codes under memoryless conditions.

Findings

MDS codes achieve minimum error probability.

Error exponents of MDS, random, and linear codes are equivalent above the critical rate.

Random and linear codes are exponentially optimal regardless of block size.

Abstract

An erasure channel with a fixed alphabet size $q$, where $q \gg 1$, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel, the error exponent of MDS codes are compared with that of random codes and linear random codes. It is shown that the envelopes of all these exponents are identical for rates above the critical rate. Noting the optimality of MDS codes, it is concluded that both random codes and linear random codes are exponentially optimal, whether the block sizes is larger or smaller than the alphabet size.