The Tight Error Linear Complexity of Periodic Sequences

Abstract

Based on linear complexity, k-error linear linear complexity, k-error linear complexity profile and minerror, the m-tight error linear complexity is presented to study the stability of the linear complexity of periodic sequences. The m-tight error linear complexity of sequence S is defined as a two tuple (k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> , LC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> ), which is the mth jump point of the k-error linear complexity profile of sequence S. The Wei-Xiao-Chen algorithm can not be generalized into a k-error linear linear complexity algorithm as it does not have a Stamp-Martin pattern. By using error vectors, an efficient approach for computing m-tight error linear complexity of binary sequences with period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is given. One of our main contributions is to show that every fast algorithm for linear complexity can be generalized to a fast algorithm for m-tight error linear complexity for small m.