Minimum Homogeneous Weights of a Class of Cyclic Codes over Primary Integer Residue Rings∗

EXTENDED ABSTRACT. Most of the results in traditional finite-field linear coding theory regarding the minimum distance of linear codes refer to the Hamming metric. Important early exceptions are given by Berlekamp’s nega-cyclic codes (cf. [1]) and Mazur’s [9] low-rate codes, both having interesting properties in terms of the Leemetric. At the beginning of the nineties of the previous century an important observation revealed the role of finite rings (cf. [11, 5]) and moreover the necessity to discuss more general weight functions (cf. [3, 4]). In fact, it appears that the most important ring class for contemporary coding theory is given by the class of (finite) Frobenius rings, and a most prominent weight for these rings is the homogeneous weight, which was first introduced by Heise and Constantinescu [3] for general integer residue rings. Galois rings, a subclass of all finite Frobenius rings, can be considered as Galois extensions of (primary) integer residue rings. More precisely, if f ∈ Zpm [x] is a basic monic polynomial of degree r, then the quotient ring Zpm [x]/(f) is a commutative chain ring of characteristic p, and its residue field is given by the finite field GF(p). Up to isomorphism, this ring depends only on the choice of p and r and not on the choice of f . It is denoted by GR(p, r) and was first described by Krull [7]. Homogeneous weights are defined by two properties that they share with the Hamming weight in finite field coding theory: given a finite ring R, the homogeneous weight of average value γ ∈ Q is a function w : R −→ Q satisfying w(0) = 0 and w(x) = w(y) for all x, y ∈ R with Rx = Ry under the additional requirement