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 most 2[(n +6)/10]. This bound is a consequence of some new conditions on the weight enumerator of a 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 code for all n ≤ 60; to show that self-dual codes with d ≥ 6 exist precisely for n ≥ 22, with d ≥ 8 exist precisely for n = 24, 32 and n ≥ 36, and with d ≥ 10 exist precisely for n ≥ 46; and to show that there are exactly eight self-dual codes of length 32 with d = 8. Several of the self-dual codes of length 34 have trivial group (this appears to be the smallest length where this can happen). © 1990 IEEE