On distances and self-dual codes over Fq[u]∕(ut)

Abstract

New metrics and distances for linear codes over the ring Fq[u]/(u) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. 1.Introduction. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which Fq[u]/(u) is an example) have been a source of many interesting properties [6, 8, 3]. Gulliver and Harada [4] found good examples of ternary codes over F3 using a particular type of Gray map. Siap and Ray-Chaudhuri in [9] established a relation between codes over Fq[u]/(u − a) and codes over Fq, which was used to obtained new codes over F3 and F5. In this paper we present a certain generalization of the method used in [4] and [9], defining a family of metrics for linear codes over Fq[u]/(u) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F2 + uF2 are more powerful than codes obtained by the so-called u—u+v condition. With these tools in hand, we study conditions for self-duality of codes over Fq[u]/(u). In [7] the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over Fq that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over Fq starting from a self-dual code over Fq[u]/(u). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over Fq[u]/(u) starting from a self-orthogonal code over Fq. Our results contain many of the properties studied by Bachoc for self-dual codes over F3 + uF3 in [1]. 2. Metric for Codes over Fq[u]/(u). We will use R(q, t) to denote the commutative ring Fq[u]/(u). The q elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the ∗The project was partially supported by Office of Research of the University of Michigan-Flint. †2000 Mathematics Subject Classification: Primary: 94B05, 94B60, Secondary: 11T71. ‡