It has been shown that all the known binary Golay complementary sequences of length \(2^m\) can be obtained by a single binary Golay complementary array of dimension m and size \(2\times 2 \times \cdots \times 2\) which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length \(2^m\) or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions.

已经证明，所有已知的长度为\(2^m\) 的二进制 Golay 互补序列都可以通过一个维度为m和大小为\(2\times 2 \times \cdots \times 2\ )可以用布尔函数表示。然而，构建长度为\(2^m\)的新二进制Golay互补序列或 Golay 互补阵列仍然是一个悬而未决的问题。在本文中，我们研究了二元或四元 Golay (Type-I) 互补阵列的 Walsh 谱分布和负谱分布。然后，还研究了二元II型互补阵列的沃尔什谱和二元III型互补阵列的负谱。最后，讨论了大小为 4 的互补阵列集合中二进制阵列的沃尔什谱。这项工作证明了上述二元和四元互补阵列只能由满足本文给出的谱值的（广义）布尔函数构建。例如，一个二元 I 型互补阵列必须对偶数m弯曲，对奇数m接近弯曲关于沃尔什谱，它必须是 negaplateaued、nega-bent 或 negalandscape 相对于 nega 谱。另一方面，新的二元和四元互补数组的构造可以帮助我们找到具有特定条件的新（广义）布尔函数，例如弯曲或负弯曲函数。