MDS codes over finite principal ideal rings

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 G R( p e , l) (including 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 G F(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.

Negacyclic self-dual codes over finite chain rings

“…If C is linear over R, then d H (C) ≤ n − k(C) + 1 (cf. [10]). A linear code C over R meeting the bound above is called a Maximal Distance with respect to Rank (MDR) code.…”

Independence of vectors in codes over rings

“…We know that principal ideal rings are Frobenius rings (see Proposition 2.9 in [1]). Let R be a principal ideal ring.…”

“…We know that the unique maximal ideal in Z 4 is generated by 2, and the unique maximal ideal is generated by 3 in Z 9 respectively, and the nilpotency indexes of 2 and 3 are both 2. The inverse images of (2, 1) and (1,3) in Z 4 × Z 9 are 10 and 21 in Z 36 . We have In order to prove that any two bases of a code over a finite principal ideal ring have the same number of vectors, we shall first prepare some lemmas and theorems.…”

Construction of MDS self-dual codes over Galois rings

“…Self-dual codes over Z 4 (e.g., [4]) have been generalized to self-dual codes over Z 2k with connections to even unimodular lattices and modular forms [1], to self-dual codes over Z 8 and Z 9 [12], to self-dual codes over Z m [11], and to self-dual codes over finite principal ideal rings [14]. We refer to the chapter on self-dual codes in the Handbook of Coding Theory [28] for a detailed description.…”

“…[16] have constructed an infinite family of self-dual quadratic double circulant codes over GR (4,2). MDS self-dual codes over GR(2 m , r) have been constructed using ReedSolomon codes [14]. Little is known about MDS self-dual codes over GR(p m , r) when p is odd and r > 1.…”