The present paper introduces a novel method for the construction of a class of sequences that have a zero-correlation zone. For the proposed sequence set, both the cross-correlation function and the side lobe of the auto-correlation function are zero for phase shifts within the zero-correlation zone. The proposed scheme can generate a set of sequences of length 8n2 from an arbitrary Hadamard matrix of order n and a set of 2n trigonometric-like function sequences of length 4n. The proposed sequence construction can generate an optimal zero-correlation zone sequence set that satisfies the theoretical bound on the number of members for the given zero-correlation zone and sequence period. The auto-correlation function of the proposed sequence is equal to zero for all nonzero phase shifts. The peak factor of the proposed sequence set is √2, and the peak factor of a single trigonometric function is equal to √2. Assigning the sequences of the proposed set to a synthetic aperture ultrasonic imaging system would improve the S/N of the obtained image. The proposed sequence set can also improve the performance of radar systems. The performance of the applications of the proposed sequence sets are evaluated.

The present paper introduces the construction of quadriphase sequences having a zero-correlation zone. For a zero-correlation zone sequence set of N sequences, each of length l, the cross-correlation function and the side lobe of the autocorrelation function of the proposed sequence set are zero for the phase shifts τ within the zero-correlation zone z, such that |τ|≤z (τ ≠ 0 for the autocorrelation function). The ratio $rac{N(z+1)}{ell}$ is theoretically limited to one. When l=N(z+1), the sequence set is called an optimal zero-correlation sequence set. The proposed zero-correlation zone sequence set can be generated from an arbitrary Hadamard matrix of order n. The length of the proposed sequence set can be extended by sequence interleaving, where m times interleaving can generate 4n sequences, each of length 2m+3n. The proposed sequence set is optimal for m=0,1 and almost optimal for m>1.

The present paper introduces a novel construction of structured ternary sequences having a zero-correlation zone (ZCZ) for both periodic and aperiodic correlation functions. The cross-correlation function and the side lobe of the auto-correlation function of the proposed sequence set are zero for phase shifts within the ZCZ. The proposed ZCZ sequence set can be generated from an arbitrary Hadamard matrix of order n. The sequence set of order 0 is identical to the r-th row of the Hadamard matrix. For m≥0, the sequence set of order (m+1) is constructed from the sequence set of order m by sequence concatenation and interleaving. The sequence set of order m has 2m subsets of size n. The length of the sequence is equal to n4m+2m+1(2m-1); The phase shift of the ZCZ for the whole sequence set is from -(2m-1) to (2m-1). The sequence set of order 0 is coincident with the rows of the given Hadamard sequence with no ZCZ. The subsets can be associated with a perfect binary tree of height m with 2m leaves. The r-th sequence subset consists of from the nr-th sequence to the ((n+1)r-1)-th sequence. The r-th subset is assigned to the r-th leaf of the perfect binary tree. For a longer distance between the corresponding leaves to the r-th and s-th sequences, the ZCZ of the r-th and s-th sequences is wider. This tree-structured width of ZCZ of a pair of the proposed sequences enables flexible design in applications of the proposed sequence set. The proposed sequence is suitable for a heterogeneous wireless network, which is one of the candidates for the fifth generation of radio access networks.

To analyze the structure of a set of high-dimensional perfect sequences over a composition algebra over R, we developed the theory of Fourier transforms of the set of such sequences. We define the discrete cosine transform and the discrete sine transform, and we show that there exists a relationship between these transforms and a convolution of sequences. By applying this property to a set of perfect sequences, we obtain a parameterization theorem. Using this theorem, we show the equivalence between the left perfectness and right perfectness of sequences. For sequences of real numbers, we obtain the parameterization without restrictions on the parameters.