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 mathbbZpe{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 ≥ 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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机译：本文的目的是研究有限主理想环上的代码。为此，我们从有限链环上的代码开始，作为伽罗瓦环GR（p e ，l）上代码的自然概括（包括mathbbZ p e {mathbb {Z} _ {p ^ e}}）。我们给出了关于有限链环上的MDS代码的存在以及有限链环上的自对偶代码的存在的充分条件。我们还针对长度≥2且长度≥2的Galois环GF（2 e ，l）构造MDS自对偶代码，其中a≥1且l≥2。引入了有限链环的剩余残基场，并推导了它们的一些性质。最后，我们通过广义的中文余数定理，通过检查其组成链环上的代码，来描述有限主理想环上的MDS代码和自对偶代码。
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