Abstract
Two codewords of a block code of length n and minimum distance d must differ on any set of n − d + 1 coordinates, since they are at distance at least d from each other. This observation leads to the Singleton bound, Theorem 6.1. A code whose parameters give an equality in the Singleton bound is called a maximum distance separable code or simply an MDS code. Therefore, an MDS code is a block code in which every possible (n − d + 1)-tuple of elements of the alphabet occurs in a unique codeword for any set of n − d + 1 coordinates. The focus in this chapter will be on linear MDS codes, since not so much is known about non-linear MDS codes, and there are no known non-linear MDS codes which outperform linear MDS codes.
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