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              <p>We characterize Product-MDS pairs of linear codes, i.e. pairs of codes C, D whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions dim C, dim D. We prove in particular, for C = D, that if the square of the code C has minimum distance at least 2, and (C, C) is a Product-MDS pair, then either C is a generalized Reed-Solomon code, or C is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics.</p>
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