Nonexistence of near-extremal formally self-dual even codes of length divisible by 8

It is a well known fact that if C is an [n, k, d] formally self-dual even code with n > 30, then d ≤ 2[n/8]. A formally self-dual even code with d = 2[n/8] is called nearextremal. Kim and Pless [9] conjecture that there does not exist a near-extremal formally self dual even (not Type II) code of length n ≥ 48 with 8|n. In this paper, we prove that if n ≥ 72 and 8|n, then there is no near-extremal formally self-dual even code. This result comes from the negative coefficients of weight enumerators. In addition, we introduce shadow transform in near-extremal formally self-dual even codes. Using this we present some results about the nonexistence of near-extremal formally self-dual even codes with n = 48, 64.