Improved estimates via exponential sums for the minimum distance of Z/sub 4/-linear trace codes
Tor Helleseth, P. Vijay Kumar, Óscar Moreno, A.G. Shanbhag · IEEE Transactions on Information Theory · 1996
An upper hound for Weil-type exponential sums over Galois rings was derived by Kumar, Helleseth, and Calderbank (see ibid., vol.41, no.3, p.456, 1995). This bound leads directly to an estimate for the minimum distance of Z/sub 4/-linear trace codes. An improved minimum-distance estimate is presented. First, McEliece's result on the divisibility of the weights of binary cyclic codes is extended to Z/sub 4/ trace codes. The divisibility result is then combined with the techniques of Serre (1983) and of Moreno and Moreno (see ibid., vol.40, no.11, p.1101, 1994) to derive the improved minimum-distance estimate. The improved estimate is tight for the Kerdock code as well as for the Delsarte-Goethals codes.