The nonexistence of near-extremal formally self-dual codes

A code C is called formally self-dual if C and C-perpendicular to have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over F-2, F-3, and F-4. These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang's systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8 vertical bar n was dealt with by Han and Lee.