Bounds on the Number of Affine, Symmetric, and Hadamard Designs and Matrices
Bounds on the Number of Affine, Symmetric, and Hadamard Designs and Matrices
复制标题
仿射、对称和 Hadamard 设计和矩阵的数量界限
DOI：
10.1006/jcta.2000.3060
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
V. Tonchev
中科院分区：
文献类型：
--
作者：
C. Lam;S. Lam;V. Tonchev
摘要：得到了对称网到具有经典参数的仿射设计、仿射设计到具有经典参数的对称设计以及阶为\(n\)的对称哈达玛设计到阶为\(2n\)的对称哈达玛设计的非同构嵌入数量的下界。容尼克尔关于包含经典\((q,q^{d - 2})\) - 网的仿射\(2-(q^{d},q^{d - 1},(q^{d - 1}-1)/(q - 1))\)设计（\(d\geqslant3\)）数量的界被改进了\(q^{3 + 4+\cdots + d}(q - 1)^{d - 2}\)倍。类似地，容尼克尔关于包含经典仿射设计\(AG(d,q)\)作为剩余设计的对称\(2 - ((q^{d + 1}-1)/(q - 1),(q^{d}-1)/(q - 1),(q^{d - 1}-1)/(q - 1))\)设计（\(d\geqslant3\)）数量的界被改进到与坎托的界一致。此外，对于较大的\(d\)，从刚性对称和仿射设计出发，非同构对称\(2 - ((q^{d + 1}-1)/(q - 1),(q^{d}-1)/(q - 1),(q^{d - 1}-1)/(q - 1))\)设计数量的下界被改进到\((q^{d - 1}+\cdots + q)!\)。通过使用阶为\(n=(q + 1)/4\)（\(q\equiv3\ (\text{mod}\ 4)\)为素数幂）的佩利设计，也得到了阶为\(q + 1\)的哈达玛设计数量的下界。特别地，通过选择非经典网和非经典仿射设计作为起点，对称\(2-(40,13,4)\)设计数量的界从\(389\)提高到\(1,108,800\)，仿射\(2-(64,16,5)\)设计数量的界从\(157\)提高到\(10,810,800\)。类似的方法还将非同构哈达玛\(2-(31,15,7)\)设计的数量从\(1,266,891\)提高到\(11,727,788\)，将非同构哈达玛\(2-(39,19,9)\)设计的数量从\(38\)提高到\(5.87\times10^{14}\)。阶为\(40\)的不等价哈达玛矩阵的数量至少为\(3.66\times10^{11}\)。
Abstract Lower bouds on the number of non-isomorphic embeddings of a symmetric net into affine designs with classical parameters, of an affine design into symmetric designs with classical parameters, and of a symmetric Hadamard design of order n into ones of order 2 n are obtained. The bound of Jungnickel on the number of affine 2-( q d , q d −1 , ( q d −1 −1)/( q −1)) designs ( d ⩾3) that contain the classical ( q , q d −2 )-net is improved by a factor of q 3+4+…+ d ( q −1) d −2 . Similarly, the bound of Jungnickel for the number of symmetric 2-(( q d +1 −1)/( q −1), ( q d −1)/( q −1), ( q d −1 −1)/( q −1)) designs ( d ⩾3) that contain the the classical affine design AG ( d , q ) as a residual design is improved to match that of Kantor. Furthermore, for d large and by starting with rigid symmetric and affine designs, the lower bound for the number of non-isomorphic symmetric 2-(( q d +1 −1)/( q −1), ( q d −1)/( q −1), ( q d −1 −1)/( q −1)) designs is improved to ( q d −1 +…+ q )!. By using the Paley design of order n =( q +1)/4, q ≡3 (mod 4) a prime power, a lower bound for the number of Hadamard designs of order q +1 is also obtained. In particular, by choosing a non-classical net and non-classical affine design as the starting point, the bound on the number of symmetric 2-(40, 13, 4) designs is improved from 389 to 1, 108, 800, and the bound on the number of affine 2-(64, 16, 5) designs is improved from 157 to 10, 810, 800. A similar method also improves the number of non-isomorphic Hadamard 2-(31, 15, 7) designs from 1, 266, 891 to 11, 727, 788 and the number of non-isomorphic Hadamard 2-(39, 19, 9) designs from 38 to 5.87×10 14 . The number of inequivalent Hadamard matrices of order 40 is at least 3.66×10 11 .