Ring of the Weight Enumerators of Triply Even Codes
A binary code is said to be triply even if the weight of each element of the code is a multiple of 8. In this talk, the ring of the weight enumerators of triply even codes containing all one vector is determined for small genera.
The main ingredient is the invariant theory of the finite group. Let be a positive integer. The weight enumerator in genus has variables on which acts naturally. We find a finite group an element of which preserves the weight enumerator of a triply even code. For small , we express the invariant ring of by the weight enumerators.