Generalized Multiplexed Sequences

Abstract

Let LSR 1, LSR2,…, LSRk and LSR be k+1 linear feedback shift registers with characteristic polynomials f1(x), f2(x),…, fk(x) and g(x) over F2 and output sequences a 1, a 2,…, a k and b respectively, where a i=(ai0, ai1,…), i=1, 2,…, k, b=(b0, b1,…). Let (Formula presented.) be the k-dimensional space over F2 and γ be an injective map from F2k into the set {0, 1, 2,…, n−1}, 2k⩽n, of course. Constructing k-dimensional vector sequence A=(A0, A1,…) where At=(a1t, a2t,…, akt), t=0, 1, 2, 3,… and applying γ to each term of the sequence A, we get the sequence γ(A)=(γ(A0), γ(A1),…) where γ(At) ∈ {0, 1,…, n−1}, for all t. Using γ(A) to scramble the output sequence b of LSR, we get the sequence u=(u0, u1,…) where (Formula Presented.), for all t. we call γ a scrambling function and u the Generalized Multiplexed Sequence (generalizing Jenning’s Multiplexed Sequence, see ref, [1]), in brief, GMS. In the present paper, the period, characteristic polynomial, minimum polynomial and translation equivalence properties of the GMS are studied under certain assumptions. Let Ω be the algebraic closure of F2. Throughout this paper, andy algebraic extension of F2 are assumed to be contained in Ω. Let f(x) and g(x) be polynomials over F2 without multiple roots. Let f*g be the monic polynomial whose roots are all the distinct elements of the set S={α·β | α, β∈Ω, f(α)=0, g(β)=0}. It is well known that f*g is polynomial over F2. Let G(f) denote the vector space consisting of all output sequences of LSR with characteristic polynomial f(x).