On the $k$-error linear complexity for $p^n$-periodic binary sequences via hypercube theory

The linear complexity and the $k$-error linear complexity of a sequence are important security measures for key stream strength. By studying sequences with minimum Hamming weight, a new tool called hypercube theory is developed for $p^n$-periodic binary sequences. In fact, hypercube theory is very important in investigating critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a general hypercube decomposition approach. Second, a characterization is presented about the first decrease in the $k$-error linear complexity for a $p^n$-periodic binary sequence $s$ based on hypercube theory. One significant benefit for the proposed hypercube theory is to construct sequences with the maximum stable $k$-error linear complexity. Finally, a counting formula for $m$-hypercubes with the same linear complexity is derived. The counting formula of $p^n$-periodic binary sequences which can be decomposed into more than one hypercube is also investigated.