14,321 research outputs found
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 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
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 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
Remarks on the k-error linear complexity of p(n)-periodic sequences
Recently the first author presented exact formulas for the number of 2ⁿn-periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k >2, of a random 2ⁿn-periodic binary sequence. A crucial role for the analysis played the Chan-Games algorithm. We use a more sophisticated generalization of the Chan-Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for pⁿn-periodic sequences over Fp, p prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of pⁿn-periodic sequences over Fp
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
- …