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The linear complexity and thek-error linear complexity of a sequence are important securi ty measures for key stream strength. By studying sequences with minimum Ham ming weight, a new tool called hypercube theory is developed forp-periodic binary sequences. In fact, hypercube theory is ve ry important in investigating critical error linear complexity spectrum proposed by Etzion et al. T o demonstrate the importance of hypercube theory, we first give a general hypercube decomposition approach. Se cond, a characterization is presented about the first decrease in thek-error linear complexity for ap-periodic binary sequence s based on hypercube theory. One significant benefit for the proposed hypercube theory is to co nstruct sequences with the maximum stable krror linear complexity. Finally, a counting formula for m-hypercubes with the same linear complexity is derived. The counting formula ofp-periodic binary sequences which can be decomposed into mor e than one hypercube is also investigated.

mathematical foundations of computing

On the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-error linear complexity for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p^n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-periodic binary sequences via hypercube theory