Berlekamp-Massey algorithm
The Berlekamp-Massey algorithm is used for finding the minimal polynomial of a linearly recurrent sequence. The algorithm itself is presented at the end of this article.
Definition 1.
Definition 2.
Given a linearly recurrent sequence , suppose with satisfy, for all
Then the polynomial
is called an annihilator for .
Proposition 1.
The annihilators of form an ideal of .
Definition 3.
Since is a principal ideal domain, the ideal of ’s annihilators have a unique monic generator of minimal degree. This annihilator is called the minimal polynomial of .
To find the minimal polynomial, we need to be given an upper bound on its degree; having done so, the minimal polynomial is uniquely determined by the first elements of (since we need to get equations to solve for the unknowns ).
There is another way to determine the minimal polynomial, originally presented by Dornstetter, which uses the Euclidean Algorithm. It can be shown that the characteristic polynomial of a sequence is the unique monic polynomial of least degree for which the infinite product
has finitely many nonzero terms. (In fact, the nonzero terms will have coefficients up to where is the degree of ).
We can rewrite this as
This is where the Euclidean Algorithm comes in; if we take the GCD of and , keeping track of remainders, we get two sequences such that
forms a series of polynomials whose degree is decreasing; as soon as this degree is less than , we have the needed polynomials with .
There is more info about the Extended Euclidean Algorithm in “Modern Computer Algebra” by von zur Gathen and Gerhard.
(Berlekamp’s algorithm proper to come)
The original algorithm is from Algebraic Coding Theory by Elwyn R. Berlekamp, McGraw-Hill, 1968. Its application to linearly recurrent sequences was noted by J.L.Massey, in “Shift-register synthesis and BCH decoding”, 1969.
Title | Berlekamp-Massey algorithm |
Canonical name | BerlekampMasseyAlgorithm |
Date of creation | 2013-03-22 14:28:55 |
Last modified on | 2013-03-22 14:28:55 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 7 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 15A03 |
Classification | msc 11B37 |
Related topic | RecurrenceRelation |
Related topic | MapleImplementationOfBerlekampMasseyAlgorithm |
Defines | linear recurrent sequence |
Defines | minimal polynomial of a sequence |
Defines | annihilator |