Multiplicities of Character Values of Binary Sidel'nikov-Lempel-Cohn-Eastman Sequences

TL;DR

This paper investigates the multiplicities of character values in binary SLCE sequences, expressing roots via Jacobi sums and deriving divisibility results using Gauss and Jacobi sums in specific cases.

Contribution

It introduces a novel approach to analyze roots of character polynomials of SLCE sequences using Jacobi sums, advancing understanding of their algebraic properties.

Findings

Expressed multiple roots of character polynomials via Jacobi sums

Derived new divisibility results for SLCE sequences in the semiprimitive case

Connected roots multiplicities with algebraic sum evaluations

Abstract

Binary Sidel'nikov-Lempel-Cohn-Eastman sequences (or SLCE sequences) over F 2 have even period and almost perfect autocorrelation. However, the evaluation of the linear complexity of these sequences is really difficult. In this paper, we continue the study of [1]. We first express the multiple roots of character polynomials of SLCE sequences into certain kinds of Jacobi sums. Then by making use of Gauss sums and Jacobi sums in the "semiprimitive" case, we derive new divisibility results for SLCE sequences.