The linear complexity and the $k$-error linear complexity of a sequence havebeen used as important security measures for key stream sequence strength inlinear feedback shift register design. By using the sieve method ofcombinatorics, the $k$-error linear complexity distribution of $2^n$-periodicbinary sequences is investigated based on Games-Chan algorithm. First, for $k=2,3$, the complete counting functions on the $k$-error linearcomplexity of $2^n$-periodic binary sequences with linear complexity less than$2^n$ are characterized. Second, for $k=3,4$, the complete counting functionson the $k$-error linear complexity of $2^n$-periodic binary sequences withlinear complexity $2^n$ are presented. Third, for $k=4,5$, the completecounting functions on the $k$-error linear complexity of $2^n$-periodic binarysequences with linear complexity less than $2^n$ are derived. As a consequenceof these results, the counting functions for the number of $2^n$-periodicbinary sequences with the 3-error linear complexity are obtained, and thecomplete counting functions on the 4-error linear complexity of $2^n$-periodicbinary sequences are obvious.
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机译：序列的线性复杂度和$ k $-误差线性复杂度已用作密钥流序列强度线性反馈移位寄存器设计的重要安全措施。利用组合筛的筛分方法，基于Games-Chan算法研究了$ 2 ^ n $周期二元序列的$ k $误差线性复杂度分布。首先，对于$ k = 2,3 $，表征了线性复杂度小于$ 2 ^ n $的$ 2 ^ n $-周期二进制序列的$ k $-误差线性复杂度的完整计数函数。其次，对于$ k = 3,4 $，给出了线性复杂度为$ 2 ^ n $的$ k $误差线性复杂度$ 2 ^ n $-周期二进制序列的完整计数函数。第三，对于$ k = 4,5 $，推导了线性复杂度小于$ 2 ^ n $的$ k $误差线性复杂度$ 2 ^ n $-周期二元序列的完全计数函数。这些结果的结果是，获得了具有3误差线性复杂度的$ 2 ^ n $周期二进制序列数的计数函数，得到了关于$ 2 ^ n $周期性二进制序列的4误差线性复杂度的完整计数函数。明显。
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