Basic properties of the complete complementary codes using the DFT matrices and the Kronecker products

In this paper we investigate fundamental properties of the (M,N,L)-complete complementary codes proposed by Suehiro and Hatori. The (M,N,L)-complete complementary code satisfies: (a) for each set, the sum of autocorrelation functions is equal to zero except for the zero shift, and (b) for any two different sets, the sum of cross-correlation functions are equal to zero at every shift. We first show that, given (M
₁
,N
₁
1, L
₁
)-and (M
₂
,N
₂
, L
₂
)-complete complementary codes, we can easily construct an (M
₁
M
₂
,N
₁
N
₂
, L
₁
L
₂
)-complete complementary code by using the Kronecker product. We also show that an (N, N,N)-complete complementary code is obtained from the N-th DFT matrix by the cyclic shifts.