A new upper bound on the minimal distance of self-dual codes

It is shown that the minimal distance d of a binary self-dual code of length n ≥ 74 is at most2 [ (n + 6 )/10 ]. This bound is a consequence of some new conditions on the weight enumerator ofa self-dual code obtained by considering a particular translate of the code called its ‘‘shadow’’.These conditions also enable us to find the highest possible minimal distance of a self-dual codefor all n ≤ 60; to show that self-dual codes with d ≥ 6 exist precisely for n ≥ 22, with d ≥ 8 existprecisely for n = 24, 32 and n ≥ 36, and with d ≥ 10 exist precisely for n ≥ 46; and to show thatthere are exactly eight self-dual codes of length 32 with d = 8. Several of the self-dual codes oflength 34 have a trivial group (this appears to be the smallest length where this can happen). ___________* This paper appeared in IEEE Trans. Inform. Theory, vol. 36 (Nov. 1990), pp. 1319-1333.