Cyclic Codes and Self-Dual Codes Over

Abstract

We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u 2 = 0. This ring shares many properties of Z 4 and F 4 and admits a linear "Gray map". Cyclic codes are described as modules over (F 2 + uF 2 ) n which may not be free. Self-dual codes of odd length exists as in the case of Z 4 -codes. We exhibit some extremal codes of this very interesting family. Index Terms: Codes over rings, Cyclic codes, Self-dual codes, Gray map. 1 Introduction Among the four rings of four elements, the Galois field F 4 and more recently the ring of integers modulo four Z 4 are the most used in coding theory. Z 4 -codes are renowned for producing good nonlinear codes by the Gray map, namely Kerdock, Preparata or Goethals codes. On the other hand, the ring F 4 admits a linear Gray map which does not give good binary codes. The ring R = F 2 + uF 2 shares some good properties of both Z 4 and F 4 . This alphabet is given by all binary polynomials in indeterminate u of degr...