On the k-Error Linear Complexities of De Bruijn Sequences

Abstract

We 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.