Walsh spectrum and nega spectrum of complementary arrays

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.