Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences

Jianqin Zhou; Wanquan Liu; Xifeng Wang

doi:10.3934/amc.2017036

Article Contents

2017, Volume 11, Issue 3: 429-444. Doi: 10.3934/amc.2017036

This issue Previous Article Parity check systems of nonlinear codes over finite commutative Frobenius rings Next Article Integer-valued Alexis sequences with large zero correlation zone

Complete characterization of the first descent point distribution for the k-error linear complexity of 2ⁿ-periodic binary sequences

1.
Department of Computing, Curtin University, Perth, WA 6102, Australia
2.
School of Computer Science, Anhui Univ. of Technology, Maanshan, Anhui 243032, China

Received: September 2014

Revised: August 2015

Published: September 2017

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Abstract

In this paper, a new constructive approach of determining the first descent point distribution for the $k$-error linear complexity of $2^n$-periodic binary sequences is developed using the sieve method and Games-Chan algorithm. First, the linear complexity for the sum of two sequences with the same linear complexity and minimum Hamming weight is completely characterized and this paves the way for the investigation of the $k$-error linear complexity. Second we derive a full representation of the first descent point spectrum for the $k$-error linear complexity. Finally, we obtain the complete counting functions on the number of $2^n$-periodic binary sequences with given $2^m$-error linear complexity and linear complexity $2^n-(2^{i_1}+2^{i_2}+···+2^{i_m})$, where $0≤ i_1<i_2<···<i_m<n. $ In summary, we depict a full picture on the first descent point of the $k$-error linear complexity for $2^n$-periodic binary sequences and this will help us construct some sequences with requirements on linear complexity and $k$-error complexity.

Keywords:

Periodic sequence,
linear complexity,
$k$-error linear complexity,
$k$-error linear complexity distribution.

Mathematics Subject Classification: Primary: 94A55, 94A60; Secondary: 11B50.

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