Optimal ternary sequence sets rigorously achieving the Welch bound

Abstract

Sequences are finite-dimensional discrete signals in the complex domain, which are widely used in communication, radar, sonar, and information security to complete the requirements for synchronization, multiple random access, channel estimation, ranging, anti-interference, and digital watermarking due to their anti-interference, stability, and easy implementation. There are close relations between sequences and mathematical objects such as cyclic Hadamard matrices, difference sets (families), tight frames, unbiased bases, and cyclic codes. Constructing sequences reaching or approximating the theoretical bounds has always been a core topic in sequence design and coding theory. In this paper, based on the Semi-Bent function and difference set, we propose a class of optimal ternary sequence sets (the elements of sequences belong to $\{0,\pm~1\}$), which is the first class of optimal sequence set strictly achieving the Welch bound since this bound was established in 1974. Compared with the binary sequence sets (the elements of sequences take values $\pm~1$), the new sequence set is more suitable for ultra-wideband communication, digital watermarking, and spectrum constrained scenarios. While compared with known ternary sequence sets asymptotically approaching two times the Welch bound, the proposed ternary sequence sets have better correlation properties.