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

Abstract

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 (f.s.d.) even code with $d=2[n/8]$ 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 $n⩾48$ with $8|n$. In this paper, we prove that if $n⩾72$ and $8|n$, 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 $n=48,64$.