On the Computation of the Linear Complexity and the k-Error Linear Complexity of Binary Sequences with Period a Power of Two

Abstract

On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two Ana SȃlȃgeanAbstract-The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period = 2 n 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 = 2 n , 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( ). We apply our algorithm to decoding and encoding binary repeated-root cyclic codes of length in linear, O( ), time and space. A previous decoding algorithm proposed by Lauder and Paterson has O( (log ) 2 ) complexity.

Linear Complexity Measures of Binary Multisequences

“…Then the joint linear complexity of S denoted by JLC(S) is defined as the least order of a linear recurrence relation over q F that S (1) , S (2) , …, S (m) satisfy simultaneously. The polynomial of minimal degree which generates a given multisequence is called its minimal connection polynomial.…”

“…Ana Salagean [1] derived an algorithm for computing an error sequence e of minimum cost for a binary sequence of period 2 n such that c e S LC   ) ( . Here we derive an extension of above algorithm to the case of multisequences.…”

Linear complexity for sequences with characteristic polynomial ƒ^v

“…It was noted by Sȃlȃgean in [3] and by Meidl in [4] that we actually do not need to have a whole period of the sequence in order to determine its linear complexity using the GamesChan algorithm. It suffices to have a number of terms greater or equal to the linear complexity, provided we still know that the sequence admits as characteristic polynomial a power of x−1 or more generally of some irreducible polynomial f .…”

Modified Berlekamp-Massey Algorithm for Approximating the k-Error Linear Complexity of Binary Sequences

“…By taking the minimum value of the linear complexity for each number of errors, the results in the tree in Figure 1 give an incomplete approximate k-error linear complexity profile as being {(0, 8), (1,9), (2, 7), (3, 5), (5, 2)}. Applying Property 1 in Section 2 and using the fact that L w H (s) (s) = 0 the full approximate k-error linear complexity profile is found: {(0, 8), (1,8), (2,7), (3,5), (4,5), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2), (11, 0)}.…”

“…One might also be interested in the minimum number of errors needed to achieve a linear complexity below a certain set value L M ax (see Sȃlȃgean [9]). This again would make some of the recursive calls unnecessary, when the current complexity is already equal to or below L M ax .…”