On the k-Error Linear Complexities of De Bruijn Sequences

Abstract

AbstractWe study the k-error linear complexities of de Bruijn sequences. Let n be a positive integer and k be an integer less than \(\lceil \frac{2^{n-1}}{n}\rceil \). We show that the k-error linear complexity of a de Bruijn sequence of order n is greater than or equal to \(2^{n-1}+1\), which implies that de Bruijn sequences have good randomness property with respect to the k-error linear complexity. We also study the compactness of some related bounds, and prove that in the case that \(n\ge 4\) and n is a power of 2, there always exists a de Bruijn of order n such that the Hamming weight of \(L(\mathbf {s})\oplus R(\mathbf {s})\) is \(\frac{2^{n-1}}{n}\), where \(L(\mathbf {s})\) and \(R(\mathbf {s})\) denote respectively the left half and right half of one period of this de Bruijn sequence. Besides, some experimental results are provided for the case that n is not a power of 2.Keywordsk-Error linear complexityde Bruijn sequenceNonlinear feedback shift register