A note on formally self-dual even codes of length divisible by 8

A binary code with the same weight distribution as its dual code is called formally self-dual ( f.s.d.). We only consider f.s.d. even codes (codes with only even weight codewords). We show that any formally self-dual even binary code of length n not divisible by 8 is balanced. We show that the weight distribution of a balanced near-extremal f.s.d. even code of length a multiple of 8 is unique. We also determine the possible weight enumerators of a near-extremal f.s.d. even [n,n/2,2 n/8 ] code with 8n as well as the dimension of its radical.