On Constructions of Regular Hadamard Matrices and Bent Functions

Using the generalized Kronecker construction from two binary Hadamard matrices, we build new families of Hadamard matrices of square order, which keep many properties of the original matrices. Specifically, when one of the initial matrices is the Sylvester Hadamard matrix, the resulting Hadamard matrix is regular and has a bent function in each row or in each column. These bent functions belong to the Maiorana–McFarland class. When both initial Hadamard matrices are Sylvester Hadamard, regular Hadamard matrices having a bent function in each row and column are obtained. In this case, we can construct bent functions with maximal algebraic degree. Moreover, we are able to compute the dimension of the kernel and an upper bound for the rank. Finally, we give two specific constructions using Latin squares.