Regular Hadamard matrix, maximum excess and SBIBD

When k = q1, q2, q1q2, q1q4, q2q3N , q3q4N , where q1, q2 and q3 are prime powers, and where q1 ≡ 1 (mod 4), q2 ≡ 3 (mod 8), q3 ≡ 5 (mod 8), q4 = 7 or 23, N = 23t, a, b = 0 or 1, t = 0 is an arbitrary integer, we prove that there exist regular Hadamard matrices of order 4k, and also there exist SBIBD(4k, 2k + k, k + k). We find new SBIBD(4k, 2k + k, k + k) for 233 values of k. ∗ The second author is supported by the NSF of China (No. 10071029). Australasian Journal of Combinatorics 27(2003), pp.263–275 1 Preliminaries An n× n matrix H is called a Hadamard matrix (or H-matrix) if every entry of the matrix is 1 or −1, and HH = nIn, where In is an n×n identity matrix. In this paper we use H to denote the transpose of a matrix H. We denote the excess of an H-matrix H = [aij ] by σ(H), where σ(H) = ∑ 1≤i,j≤n aij . Let σ(n) = max{σ(H)}. The weight of an H-matrix H, denoted by W (H), is the number of ones in H. We define W (n) = max{W (H)}. Note that the maxima are taken over all n × n H-matrices H. It is obvious that σ(H) = 2W (H) − n and σ(n) = 2W (n)− n (see [4], [5], [6], [7] for details). Best [1] proved that σ(n) ≤ n√n. (1) Definition 1 (Regular Hadamard Matrix) A regular Hadamard matrix has the sum of each column of the matrix and the sum of each row of the matrix constant. Definition 2 (SBIBD) A symmetric balanced incomplete block design, called an SBIBD(v, k, λ), is defined by a v × v matrix M , which has every entry 0 or 1. The sum of each column and the sum of each row of the matrix is k. For any two columns ci, cj (and two rows ri, rj), 1 ≤ i = j ≤ v, the inner product of ci and cj (ri and rj) is λ (see [10]). With the result of this paper and those of [4], [9], the status of the existence of 4k-Hadamard matrices and SBIBD(4k, 2k + k, k + k) is that they exist for k ∈ {1, 3, 5, · · ·, 45, 49, · · ·, 69, 73, 75, 81, · · ·, 101, 105, 107, 109, · · ·, 125, 129, 131, 135, 137, 139, 143, · · ·, 149, 153, · · ·, 165, 169, · · ·, 175, · · ·, 189, 193, · · ·, 197, 201, · · ·, 207, 211, 215, 219, 221, 225, 227, 229, 233, 235, 241, · · ·, 251, 257, 259, 261, 267, 269, 273, 275, 277, 281, · · ·, 299, 303, 307, 313, · · ·, 327, 331, · · ·, 339, 343, · · ·, 353, 361, 363, 371, 373, 375, 379, 387, 389, 391, 393, 397, 401, 405, · · ·, 411, 415, 417, 419, 421, 427, 429, 433, 441, 443, 447, 449, 451, 457, 461, 467, 471, 475, 477, 489, 491, 495, 499, 507, 509, 511, 513, 519, 521, 523, 525, 529, 531, · · ·, 543, 547, 549, 551, 557, 559, 563, 567, 569, 571, 575, 577, 579, 583, 587, 591, 593, 601, 603, 605, 609, 613, 617, · · ·, 625, 633, 637, 641, 643, 645, 653, 655, 659, 661, 667, 671, 673, 675, 677, 679, 683, 687, 691, 695, 699, 701, 703, 707, 709, 723, 725, 729, 731, 733, 735, 739, 741, 747, 753, 757, 761, 763, 767, 769, 771, 773, 777, 779, 783, 787, 791, 797, 803, 807, 809, 811, 815, 819, 821, 827, 829, 831, 841, · · ·, 859, 865, 867, 871, 875, 877, 879, 881, 883, 885, 891, 895, 897, 907, 909, 921, 925, 929, 931, 937, 939, 941, · · ·, 947, 951, 953, 957, 959, 961, 963, 971, 975, 977, 979, 981, 993, 997, 999, q1, q2, q1q2, q1q4, q2q3N , q3q4N}, where q1, q2 and q3 are prime powers, q1 ≡ 1 (mod 4),