Trace representation of a family of generalized cyclotomic binary sequences with period \(p^n\)

The article presents a comprehensive study on the trace representation of a family of generalized cyclotomic binary sequences with a specific period. It begins by highlighting the importance of pseudorandom sequences in various engineering and scientific applications, emphasizing the need for sequences with long periods, low autocorrelation, and large linear complexity. The paper then introduces the concept of cyclotomy as a tool for constructing sequences with good randomness properties, discussing several known sequences and their optimal autocorrelation properties. The main focus is on a new family of balanced binary sequences, initially proposed by Xiao et al. and extended by Edemskiy et al., which are constructed using a novel generalized cyclotomy. The article employs the discrete Fourier transform to compute the trace representation of these sequences, providing explicit formulas and detailed proofs. It also derives a formula for the linear complexity of these sequences using Blahut’s theorem, confirming existing results. The paper concludes with examples illustrating the trace representation and linear complexity calculations, demonstrating the practical applicability of the theoretical findings.