Linear complexity of quaternary sequences over Z_4 derived from generalized cyclotomic classes modulo 2p

TL;DR

This paper precisely determines the linear complexity of certain quaternary sequences over Z_4 derived from generalized cyclotomic classes modulo 2p, addressing a complex open problem in sequence analysis.

Contribution

It provides the exact linear complexity values for these sequences using Galois rings, solving an open problem and extending understanding beyond finite field cases.

Findings

Exact linear complexity values derived from Galois ring theory

Addresses the open problem posed by Kim, Hong, and Song

Highlights the complexity of polynomial roots in Z_4[X] due to zero divisors

Abstract

We determine the exact values of the linear complexity of 2p-periodic quaternary sequences over Z_4 (the residue class ring modulo 4) defined from the generalized cyclotomic classes modulo 2p in terms of the theory of of Galois rings of characteristic 4, where p is an odd prime. Compared to the case of quaternary sequences over the finite field of order 4, it is more dificult and complicated to consider the roots of polynomials in Z_4[X] due to the zero divisors in Z_4 and hence brings some interesting twists. We answer an open problem proposed by Kim, Hong and Song.