For any integer k ≥ 1, an isometry between codes over Z 2 k+1 and codes over Z 4 is defined and used to give an equivalent generalization of the Gray map to the one introduced in [C. Carlet, IEEE Trans. Inform. Theory, 44 (1998), pp. 1543-1547. Several results related to the linearity or nonlinearity of codes over Z 2 k+1 , as well as its corresponding images under this map, are given. These results are similar to those presented in Theorems 4, 5, and 6 of [1. Introduction. For a given integer n ≥ 1 the Gray map is a bijective function from Z n 4 into F 2n 2 , and its main quality is that it is an isometry; in other words, it is a distance preserving function with respect to the Lee and Hamming distances. With this property in mind, the Gray map has been used extensively to construct binary codes from quaternary codes (i.e., codes over Z 4 ). In [4] the authors introduce a new kind of binary code called a Z 4 -linear code, establishing that a binary code of even length is Z 4 -linear if its coordinates can be arranged so that it is the image under the Gray map of a quaternary linear code. Furthermore, the authors give necessary and sufficient conditions for a binary code to be Z 4 -linear, as well as conditions under which the binary Gray map image of a quaternary linear code is a linear code. On the other hand, in [2] the author introduces a generalization of the Gray map and extends the concept of Z 4 -linear codes to Z 2 k -linear codes in a natural way. He also studies the conditions under which the binary codes will be Z 2 k -linear; in particular, a characterization of Z 8 -linear codes is given.In this paper, for k ≥ 1, an isometry between codes over Z 2 k+1 and codes over Z 4 is introduced and used together with the usual Gray map to give a generalization of the latter mapping. This generalization of the Gray map is equivalent to the one given in [2], but with this reexpression in terms of the isometry from Z n 2 k+1 to Z 2 k−1 n

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It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their weight distributions. On the other hand, it is also well known that cyclic codes with few weights have a great practical importance in coding theory and cryptography. In particular, cyclic codes having three nonzero weights have been studied by several authors, however, most of these efforts focused on cyclic codes over a prime field. In this work we present a characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field. The codes under this characterization are, indeed, optimal in the sense that their lengths reach the Griesmer lower bound for linear codes. Consequently, these codes reach, simultaneously, the best possible coding capacity, and also the best possible capabilities of error detection and correction for linear codes. But because they are cyclic in nature, they also possess a rich algebraic structure that can be utilized in a variety of ways, particularly, in the design of very efficient coding and decoding algorithms. What is also worth pointing out, is the simplicity of the necessary and sufficient numerical conditions that characterize our class of optimal three-weight cyclic codes. As we already pointed out, it is a hard problem to find this kind of characterizations. However, for this particular case the fundamental tool that allowed us to find our characterization was the characterization for all two-weight irreducible cyclic codes that was introduced by B. Schmidt and C. White (2002). Lastly, another feature about the codes in this class, is that their duals seem to have always the same parameters as the best known linear codes.

We introduce new classes of 2-weight cyclic codes which are direct sums of 1-weight irreducible cyclic codes

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