The Asymptotic Normalized Linear Complexity of Multisequences

TL;DR

This paper characterizes the asymptotic normalized linear complexity of multisequences over finite fields, establishing bounds and showing the existence of multisequences for all admissible limit pairs, thus solving an open problem.

Contribution

It provides a complete characterization of the asymptotic linear complexity bounds for multisequences, answering an open problem in the field.

Findings

Established bounds for liminf and limsup of normalized linear complexity.

Proved the existence of multisequences for all admissible limit pairs.

Solved an open problem posed by Dai, Imamura, and Yang.

Abstract

We show that the asymptotic linear complexity of a multisequence a in F_q^\infty that is I := liminf L_a(n)/n and S := limsup L_a(n)/n satisfy the inequalities M/(M+1) <= S <= 1 and M(1-S) <= I <= 1-S/M, if all M sequences have nonzero discrepancy infinitely often, and all pairs (I,S) satisfying these conditions are met by 2^{\aleph_0} multisequences a. This answers an Open Problem by Dai, Imamura, and Yang. Keywords: Linear complexity, multisequence, Battery Discharge Model, isometry.