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MDS codes over finite principal ideal rings. (English) Zbl 1178.94221

Summary: The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings \(GR(p ^{e }, l)\) (including \({\mathbb{Z}_{p^e}}\)). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings \(GF(2^{e }, l)\) of length \(n = 2^{l }\) for any \(a \geq 1\) and \(l \geq 2\). Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

MSC:

94B05 Linear codes (general theory)
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