Abstract
Self-dual codes constitute the most fascinating class of codes, by their many connections to invariant theory, combinatorial designs, and Euclidean lattices. In this short chapter, we focus our attention on existence conditions for alphabets having the structure of, successively, chain rings, commutative Frobenius rings, and noncommutative Frobenius rings. Necessary congruence existence conditions bearing on the size of the residue field are given. Sufficient existence conditions are derived in short lengths and extended to longer lengths by taking direct sums of codes.
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