# The \(k\)-error linear complexity distribution for \(2^n\)-periodic binary sequences

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## Abstract

The linear complexity and the \(k\)-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, we investigate the \(k\)-error linear complexity distribution of \(2^n\)-periodic binary sequences in this paper based on Games–Chan algorithm. First, for \(k=2,3\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences (with linear complexity less than \(2^n\)) are characterized. Second, for \(k=3,4\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences with linear complexity \(2^n\) are presented. Third, as a consequence of these results, the counting functions for the number of \(2^n\)-periodic binary sequences with the \(k\)-error linear complexity for \(k = 2\) and \(3\) are obtained.

## Keywords

Periodic sequence Linear complexity \(k\)-error linear complexity \(k\)-error linear complexity distribution## Mathematics Subject Classification (2000)

94A55 94A60 11B50## Notes

### Acknowledgments

The research was supported by Anhui Natural Science Foundation (No.1208085MF106).

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