Abstract
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 $d_{48}^+$ cannot be ganerated by the weight enumerator of self-dual doubly even code with code length n=8, 16, 24, 32 and 40.
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