On the $k$-Error Linear Complexity of $p^{m}$-Periodic Binary Sequences

Abstract

In this correspondence, we study the statistical stability properties of p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> -periodic binary sequences in terms of their linear complexity and k-error linear complexity, where p is n prime number and 2 is a primitive root modulo p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . We show that their linear complexity and k-error linear complexity take a value only from some specific ranges. We then present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity in a new viewpoint different from the approach by Meidl. We also derive the distribution of p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> -periodic binary sequences with specific k-error linear complexity. Finally, we get an explicit formula for the expectation value of the k-error linear complexity and give its lower and upper bounds, when k les [p/2].