Key points are not available for this paper at this time.
Invariant theory can be combined with coding theory to analyze the polynomial ring over its weight enumerator. For a certain class code, especially for self-dual doubly even code or type II code, the polynomials over its code satisfy the invarian condition. In this paper, we will show that the ring of the weight enumerator over integer of self-dual doubly even code d₄₈^+ cannot be ganerated by the weight enumerator of self-dual doubly even code with code length n=8, 16, 24, 32 and 40.
Azaliyah et al. (Wed,) studied this question.