Modular Sequences and Modular Hadamard Matrices

For every n divisible by 4, we construct a square matrix H of size n, with coeecients 1, such that H H t nI mod 32. This solves the 32-modular version of the classical Hadamard conjecture. We also determine the set of lengths of 16-modular Golay sequences. 0. Introduction Hadamard matrices can be constructed from various binary sequences, either from Golay complementary sequences, or Williamson quadruples or even from simple quadruples of binary sequences with vanishing correlations. These constructions (which will be recalled below as we need them) may of course be interpreted modulo some integer m > 1 and produce modular Hadamard matrices. Given a modulus m > 1, an m-modular Hadamard matrix of size n is an n n square matrix H with entries 1, such that the dot product of any two distinct rows is congruent to 0 modulo m, i.e. such that H H t nI mod m. (We let I denote the identity matrix of the appropriate size and X t is the transpose of the matrix X.) Modular Hadamard matrices have been introduced in 1972 by O. Marrero and A.T. Butson ((MB1], MB2]). It is not diicult to see that if the modulus m is divisible by 4, then the size n 3 of an m-modular Hadamard matrix must be divisible by 4, as is the case for ordinary Hadamard matrices. It is therefore natural to consider the m-modular version of the celebrated Hadamard conjecture, namely the question: Given m, does there exist an m-modular Hadamard matrix of size 1 n for every n 0 mod 4 ? In their papers, O. Marrero and A.T. Butson considered mostly cases where m is odd or m 2 mod 4. One of their results states that 6-modular Hadamard matrices exist for every even size. In this paper, we investigate moduli m of the form m = 2 e , and the special case m = 48. Our main result is perhaps that Hadamard matrices mod 32 can be obtained in any size n divisible by 4. In order to get this result we had to appeal to all 3 kinds of binary sequences: Golay pairs, Williamson quadruples, and complementary quadruples, which we propose to name Golay quadruples, and which yield Hadamard matrices by the remarkable construction of Goethals and Seidel GS]. Modular Golay pairs modulo 16 already appeared in EKS]. As stated in Proposition (7.9) of …