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An erasure channel with a fixed alphabet size $q$, where $q \gg 1$, isstudied. 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 andlinear random codes. It is shown that the envelopes of all these exponents areidentical for rates above the critical rate. Noting the optimality of MDScodes, it is concluded that both random codes and linear random codes areexponentially optimal, whether the block sizes is larger or smaller than thealphabet size.

Optimal Coding for the Erasure Channel with Arbitrary Alphabet Size