New families of Hadamard matrices with maximum excess

In this paper, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. In particular, we show that if either $4m^2+4m+3$ or $2m^2+2m+1$ is a prime power, then there exists a biregular Hadamard matrix of order $n=4(m^2+m+1)$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to be regular in terms of four-class translation association schemes on finite fields.Comment: 24 page

Regular Hadamard matrix, maximum excess and SBIBD

When k = q1, q2, q1q2, q1q4, q2q3N, q3q4N, whereq1, q2 and q3 are prime powers, and where q1 ≡ 1(mod4),q2 ≡ 3(mod8),q3 ≡ 5(mod8),q4 =7 or 23, N =2 a 3 b t 2, a, b = 0 or 1, t = 0 is an arbitrary integer, we prove that there exist regular Hadamard matrices of order 4k 2, and also there exist SBIBD(4k 2, 2k 2 + k, k 2 + k). We find new SBIBD(4k 2, 2k 2 + k, k 2 + k) for 233 values of k. The second author is supported by the NSF of China (No. 10071029)