12 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
Reducing the calculation of the linear complexity of binary u2^v-periodic sequences to Games-Chan algorithm
We show that the linear complexity of a -periodic binary sequence, odd, can easily be calculated from the linear complexities of certain -periodic binary sequences. Since the linear complexity of a -periodic binary sequence can efficiently be calculated with the Games-Chan algorithm, this leads to
attractive procedures for the determination of the linear complexity of a -periodic binary sequence. Realizations are presented for
Quadratic functions with prescribed spectra
We study a class of quadratic p-ary functions Fp,n from \F_p^n to F_p, p ≥ 2,
which are well-known to have plateaued Walsh spectrum; i.e., for each b ∈ F_p^n the Walsh transform fˆ(b) satisfies |fˆ(b)|^2 ∈ {0, p^(n+s)} for some integer 0 ≤ s ≤ n − 1. For various types of integers n, we determine possible values of s, construct Fp,n with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279–295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286–4298, 2005) on semi-bent functions, and of Fitzgerald
(Finite Fields Appl 15, 69–81, 2009) on quadratic forms
An empirical assessment of the cross-national measurement validity of graded paired comparisons
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