Linear complexity and trace representation of quaternary sequences over ℤ 4 $\mathbb {Z}_{4}$ based on generalized cyclotomic classes modulo pq

Abstract

We determine the exact values of the linear complexity of 2pperiodic 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 difficult 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 prove the main results as follows Linear Complexity =

On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation

“…We mention that it is also interesting to consider the linear complexity of quaternary sequences over Z 4 . For example, in [8], Edemskiy derived the linear complexity of quaternary sequences with optimal autocorrelation value over the finite ring Z 4 and in [3], Chen et al determined the exact values of the linear complexity of quaternary sequence over Z 4 defined from the generalized cyclotomic classes modulo 2p. In [4], Chen et al defined a family of quaternary sequences over Z 4 of length pq, a product of two distinct odd primes, using the generalized cyclotomic classes modulo pq and calculate the discrete Fourier transform of the sequences.…”

Linear Complexity of Quaternary Sequences over Z₄ Based on Ding-Helleseth Generalized Cyclotomic Classes

“…Lemma 1. Let γ ∈ GR(4, 4 ℓ ) be a primitive pq-th root of unity, then we have (1). 1 + γ p + γ 2p + .…”

“…Very recently, as a generalization of Jacobi sequences, a family of quaternary sequences over Z 4 with period pq was proposed by Edemskiy [9] and Chen [1], respectively. They determined the linear complexity with different methods.…”

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“…Most references have concentrated on the linear complexity of quaternary sequences over F 4 [4][5][6][7]. However, there has been less attention to the linear complexity of sequences over Z 4 due to the phenomenon of zero divisors in Z 4 [8].…”