12,635 research outputs found
How to determine linear complexity and -error linear complexity in some classes of linear recurring sequences
Several fast algorithms for the determination of the linear complexity of -periodic sequences over a finite
field \F_q, i.e. sequences with characteristic polynomial , have been proposed in the literature.
In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic
polynomial for an arbitrary positive integer , and are presented.
The result is then utilized to establish a fast algorithm for determining the -error linear complexity of
binary sequences with characteristic polynomial
On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two
The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period â„“ = 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 â„“ = 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(â„“). We apply our algorithm to
decoding and encoding binary repeated-root cyclic codes of length â„“ in linear, O(â„“), time and space. A previous decoding algorithm proposed
by Lauder and Paterson has O(â„“(logâ„“)2) complexity
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