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

It is a well-known fact that if CC is an [n,k,d][n,k,d] formally self-dual even code with n>30n>30, then d⩽2[n/8]d⩽2[n/8]. A formally self-dual (f.s.d.) even code with d=2[n/8]d=2[n/8] is called near-extremal. Kim and Pless [A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., available online 13 October 2005.] conjecture that there does not exist a near-extremal f.s.d. (not Type II) code of length n⩾48n⩾48 with 8|n8|n. In this paper, we prove that if n⩾72n⩾72 and 8|n8|n, then there is no near-extremal f.s.d. even code. This result comes from the negative coefficients of weight enumerators. In addition, we introduce shadow transform in near-extremal f.s.d. even codes. Using this we present some results about the nonexistence of near-extremal f.s.d. even codes with n=48,64n=48,64.