Linear complexity of quaternary sequences over Z_4 derived from generalized cyclotomic classes modulo 2p
This work addresses a theoretical challenge in sequence design for cryptography or coding theory, but it appears incremental as it extends known methods to a more complex ring structure.
The paper tackled the problem of determining the linear complexity of 2p-periodic quaternary sequences over Z_4 derived from generalized cyclotomic classes, and it provided exact values for this complexity, solving an open problem in the field.
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.