The biweight enumerator and the subconstituent algebra of the n-cube

Let C be a binary code of length n. For a four-tuple α = (α0, α1, α2, α3) of non-negative integers summing to n, let `α = ∣∣{(c, c) ∈ C × C : wt(c) = α2 + α3, ∂(c, c ) = α1 + α3,wt(c ) = α1 + α2 }∣∣ . We study the biweight enumerator WC(y0, y1, y2, y3) = ∑ α=(α0,α1,α2,α3) `α y α0 0 y α1 1 y α2 2 y α3 3 . For binary linear codes, MacWilliams identities for this enumerator were given by MacWilliams, et al. in 1972 [4]. We give a new proof of these identities using the Terwilliger algebra of the n-cube. Further, we consider some families of non-linear codes whose dual biweight enumerators necssarily have non-negative coefficients. For unrestricted codes, a characterization of the positive semidefinite cone of the Terwilliger algebra is given which leads to new inequalities for the coefficients of the biweight enumerator of an unrestricted code. Each extremal ray of the positive semidefinite cone corresponds to a projection onto some irreducible module of the Sn action on the free real vector space over the binary n-tuples. It remains to find computationally useful formulae for these inequalities.