On maximum distance separable codes over alphabets of arbitrary size

The well-known Singleton bound states that the cardinality of a code of length n with minimum distance d over a q-ary alphabet is at most q^n-d+1. Codes meeting the Singleton bound with equality are called maximum distance separable codes, or MDS codes for short. MDS codes enjoy many remarkable properties which only are proved (MacWilliams and Sloane, 1977) if the alphabet has the structure of a finite field; in particular, q should be the power of a prime. The aim of this paper is to show that many of these properties in fact hold for MDS codes over alphabets of arbitrary size. We do so without describing MDS codes as orthogonal arrays