Hadamard matrices of order 36 with automorphisms of order 17
Hadamard matrices of order 36 with automorphisms of order 17
复制标题
具有 17 阶自同构的 36 阶 Hadamard 矩阵
DOI：
10.1017/s002776300002273x
复制
发表时间：
1983
影响因子：
0.8
通讯作者：
V. Tonchev
中科院分区：
文献类型：
--
作者：
V. Tonchev
n阶哈达玛矩阵是一个n×n的由1和 - 1组成的矩阵，满足\(HH^T = nI\)。在此类矩阵中，n必定是1、2或者4的倍数。两个哈达玛矩阵\(H_1\)和\(H_2\)被称为等价的，如果存在单项矩阵\(P\)、\(Q\)使得\(PH_1Q = H_2\)。哈达玛矩阵\(H\)的一个自同构是该矩阵与其自身的一种等价关系，即一对单项矩阵\((P, Q)\)使得\(PHQ = H\)。换句话说，\(H\)的一个自同构是对其行进行置换，然后某些行乘以 - 1，这会导致其列的重新排序以及某些列乘以 - 1。所有自同构的集合在复合运算下构成一个群，称为\(H\)的自同构群（\(Aut H\)）。关于哈达玛矩阵的基本性质和应用的详细研究，例如可参见[1]、[7，第14章]、[8]。
A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HHt − nI. In such a matrix n is necessarily 1, 2 or a multiple of 4. Two Hadamard matrices H 1 and H 2 are called equivalent if there exist monomial matrices P, Q with PH 1 Q = H 2. An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair (P, Q) of monomial matrices such that PHQ = H. In other words, an automorphism of H is a permutation of its rows followed by multiplication of some rows by − 1, which leads to reordering of its columns and multiplication of some columns by − 1. The set of all automorphisms form a group under composition called the automorphism group (Aut H) of H. For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [1], [7, Chap. 14], [8].