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On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two

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journal contribution
posted on 21.08.2006, 16:18 by Ana SalageanAna Salagean
The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period ℓ = 2n requires the knowledge of the full sequence, while the quadratic Berlekamp-Massey algorithm only requires knowledge of 2c(s) terms. We show that we can modify the Games-Chan algorithm so that it computes the complexity in linear time knowing only 2c(s) terms. The algorithms of Stamp-Martin and Lauder-Paterson can also be modified, without loss of efficiency, to compute analogues of the k-error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two. We also develop an algorithm which, given a constant c and an infinite binary sequence s with period ℓ = 2n, computes the minimum number k of errors (and the associated error sequence) needed over a period of s for bringing the linear complexity of s below c. The algorithm has a time and space bit complexity of O(ℓ). We apply our algorithm to decoding and encoding binary repeated-root cyclic codes of length ℓ in linear, O(ℓ), time and space. A previous decoding algorithm proposed by Lauder and Paterson has O(ℓ(logℓ)2) complexity.

History

School

  • Science

Department

  • Computer Science

Pages

126669 bytes

Citation

SALAGEAN, A.M., 2005. On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two. IEEE transactions on information theory 51 (3) ,pp. 1145-1150

Publisher

© IEEE Transactions on Information Theory Society

Publication date

2005

Notes

This article was published in the journal, [© IEEE transactions on information theory society] and is also available at: http://ieeexplore.ieee.org/servlet/opac?punumber=18

ISSN

0018-9448

Language

en