Higher weights and graded rings for binary self-dual codes Original Research Article

The theory of higher weights is applied to binary self-dual codes. Bounds are given for the second minimum higher weight and a Gleason-type theorem is derived for the second higher weight enumerator. The second weight enumerator is shown to be unique for the putative [72,36,16] Type II code and the first three minimum weights are computed for optimal codes of length less than 32. We also determine the structures of the graded rings associated with the code polynomials of higher weights for small genera, one of which has the property that it is not Cohen–Macaulay.