Extremal weight enumerators and ultraspherical polynomials Original Research Article

We establish an upper bound for the minimum distance of a divisible code in terms of its dual distance. The bound generalizes the Mallows–Sloane bounds for self-dual codes. We obtain a linear recurrence for the distance distribution components of codes that attain the bound. From this we derive known conditions for the existence of extremal self-dual codes in a much simpler way. In the second half of the paper, we determine zeta functions for the codes that attain our new bound. Zeta functions for linear codes are defined in Duursma (Trans. Amer. Math. Soc. 351(9) (1999) 3609). Using properties of ultraspherical polynomials, we show that the zeta function of a quaternary extremal self-dual code has its zeros on the circle |T|=q−1/2 in analogy with the zeta function of an algebraic curve.