Generalized Hadamard matrices
Generalized Hadamard matrices
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DOI：
10.1090/s0002-9939-1962-0142557-0
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发表时间：
1962-06
影响因子：
4.6
通讯作者：
Hadamard Matrices;A. T. Butson;¿-i y-i-¿-i-y-i-2228698759
中科院分区：
文献类型：
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作者：
Hadamard Matrices;A. T. Butson;¿-i y-i-¿-i-y-i-2228698759
1.导论.一个所有元素都是p次单位根的h阶方阵H称为Hadamard矩阵（H（p，h）矩阵），如果HHCT= hI。已知H（2，h）矩阵仅对值h= 2和h= 4 t存在，其中t是正整数。虽然已经证明了H（2，4 t）矩阵对所有正整数t都存在，但它们的存在性仅对以下h值成立[1; 3; 4; 5; 6; 7]，其中q表示奇素数：（1.1）h= 2k;（1.2）h= qk+ 1= O（mod 4）;（1.3）h= hi（q7 c + 1）其中h1> 2是H（2，h）矩阵的阶;（1.4）h= h*（h*-1）其中h* 是形式数的乘积（1.1）和（1.2）;（1.5）h= 172;（1.6）h= h*（h*+ 3）其中h* 和h*+ 4都是形式数的乘积（1.1）和（1.2）;（1.7）h= h1 h2（qk+ 1）qk其中h1> 2，h2> 2是H（2，h）矩阵的阶;（1.8）h= hih 2s（s+ 3）其中h1> 2，h2> 2是H（2，h）矩阵的阶，并且其中s和s+ 4都具有qk+1的形式;（1.9）h=（r+ 1）2其中r和r+ 2都是素数或素数幂;（1.10）h是形式（1.1）-（1.9）的数的乘积。这张专辑是从[2]开始的。
1. Introduction. A square matrix H of order h all of whose elements are pth roots of unity is called a Hadamard matrix (H (p, h) matrix) if HHCT= hI. It is known [4] that H (2, h) matrices can exist only for values h= 2 and h= 4t, where t is a positive integer. Although it has been conjectured that H (2, 4t) matrices exist for all positive integers t, their existence has been established [1; 3; 4; 5; 6; 7] for only the following values of h, where q denotes an odd prime:(1.1) h= 2k;(1.2) h= qk+ 1= O (mod 4);(1.3) h= hi (q7c+ 1) where h1> 2 is the order of an H (2, h) matrix;(1.4) h= h*(h*-1) where h* is a product of numbers of forms (1.1) and (1.2);(1.5) h= 172;(1.6) h= h*(h*+ 3) where h* and h*+ 4 both are products of numbers of forms (1.1) and (1.2);(1.7) h= h1h2 (qk+ 1) qk where h1> 2, h2> 2 are orders of H (2, h) matrices;(1.8) h= hih2s (s+ 3) where hi> 2, h2> 2 are orders of H (2, h) matrices and where s and s+ 4 both are of the form qk+ l;(1.9) h=(r+ 1) 2 where both r and r+ 2 are prime or prime powers;(1.10) h is a product of numbers of the forms (1.1)-(1.9). This list is taken from [2].