Abstract
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(2e, l) of length n = 2l for any a ≥ 1 and l ≥ 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.
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Communicated by J. D. Key.
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Dougherty, S.T., Kim, JL. & Kulosman, H. MDS codes over finite principal ideal rings. Des. Codes Cryptogr. 50, 77–92 (2009). https://doi.org/10.1007/s10623-008-9215-5
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DOI: https://doi.org/10.1007/s10623-008-9215-5