Lattices in real quadratic fields and associated theta series arising from codes over F4 and F2 x F2

Let K = Q(root d) with d > 0 square-free and let O-K denote the ring of integers of K. Let C subset of R-n be a linear code whereRis F-4 if d = 5 (mod 8) and F-2 x F-2 if d = 1 (mod 8). One has a surjective ring homomorphism rho : O-K(n) -> R-n given by reductionmodulo (2O(K))(n). The inverse image Lambda(C) := rho(-1) (C) is a lattice associated to the code C. One can associate to Lambda C a theta series Theta Lambda(d) (C). In this paper we consider howthe theta series varies as one varies the value d. In particular, we show that for d, d ' with d > d ', one has Theta(Lambda d) (C) = Theta Lambda(d ') (C)+ O ( q d '+1\2).