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Berlekamp–Massey algorithm

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The Berlekamp–Massey algorithm is an algorithm that will find the shortest linear-feedback shift register (LFSR) for a given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse.[1] Reeds and Sloane offer an extension to handle a ring.[2]

Elwyn Berlekamp invented an algorithm for decoding Bose–Chaudhuri–Hocquenghem (BCH) codes.[3][4] James Massey recognized its application to linear feedback shift registers and simplified the algorithm.[5][6] Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm),[7] but it is now known as the Berlekamp–Massey algorithm.

Description of algorithm[edit]

The Berlekamp–Massey algorithm is an alternative to the Reed–Solomon Peterson decoder for solving the set of linear equations. It can be summarized as finding the coefficients Λj of a polynomial Λ(x) so that for all positions i in an input stream S:

In the code examples below, C(x) is a potential instance of Λ(x). The error locator polynomial C(x) for L errors is defined as:

or reversed:

The goal of the algorithm is to determine the minimal degree L and C(x) which results in all syndromes

being equal to 0:

Algorithm: C(x) is initialized to 1, L is the current number of assumed errors, and initialized to zero. N is the total number of syndromes. n is used as the main iterator and to index the syndromes from 0 to N−1. B(x) is a copy of the last C(x) since L was updated and initialized to 1. b is a copy of the last discrepancy d (explained below) since L was updated and initialized to 1. m is the number of iterations since L, B(x), and b were updated and initialized to 1.

Each iteration of the algorithm calculates a discrepancy d. At iteration k this would be:

If d is zero, the algorithm assumes that C(x) and L are correct for the moment, increments m, and continues.

If d is not zero, the algorithm adjusts C(x) so that a recalculation of d would be zero:

The xm term shifts B(x) so it follows the syndromes corresponding to b. If the previous update of L occurred on iteration j, then m = kj, and a recalculated discrepancy would be:

This would change a recalculated discrepancy to:

The algorithm also needs to increase L (number of errors) as needed. If L equals the actual number of errors, then during the iteration process, the discrepancies will become zero before n becomes greater than or equal to 2L. Otherwise L is updated and algorithm will update B(x), b, increase L, and reset m = 1. The formula L = (n + 1 − L) limits L to the number of available syndromes used to calculate discrepancies, and also handles the case where L increases by more than 1.


The algorithm from Massey (1969, p. 124) for an arbitrary field:

polynomial(field K) s(x) = ... /* coeffs are s_j; output sequence as N-1 degree polynomial) */
/* connection polynomial */
polynomial(field K) C(x) = 1;  /* coeffs are c_j */
polynomial(field K) B(x) = 1;
int L = 0;
int m = 1;
field K b = 1;
int n;

/* steps 2. and 6. */
for (n = 0; n < N; n++) {
    /* step 2. calculate discrepancy */
    field K d = s_n + ;

    if (d == 0) {
        /* step 3. discrepancy is zero; annihilation continues */
        m = m + 1;
    } else if (2 * L <= n) {
        /* step 5. */
        /* temporary copy of C(x) */
        polynomial(field K) T(x) = C(x);

        C(x) = C(x) - d  B(x);
        L = n + 1 - L;
        B(x) = T(x);
        b = d;
        m = 1;
    } else {
        /* step 4. */
        C(x) = C(x) - d  B(x);
        m = m + 1;
return L;

In the case of binary GF(2) BCH code, the discrepancy d will be zero on all odd steps, so a check can be added to avoid calculating it.

/* ... */
for (n = 0; n < N; n++) {
    /* if odd step number, discrepancy == 0, no need to calculate it */
    if ((n&1) != 0) {
        m = m + 1;
/* ... */

See also[edit]


  1. ^ Reeds & Sloane 1985, p. 2
  2. ^ Reeds, J. A.; Sloane, N. J. A. (1985), "Shift-Register Synthesis (Modulo n)" (PDF), SIAM Journal on Computing, 14 (3): 505–513, CiteSeerX, doi:10.1137/0214038
  3. ^ Berlekamp, Elwyn R. (1967), Nonbinary BCH decoding, International Symposium on Information Theory, San Remo, Italy{{citation}}: CS1 maint: location missing publisher (link)
  4. ^ Berlekamp, Elwyn R. (1984) [1968], Algebraic Coding Theory (Revised ed.), Laguna Hills, CA: Aegean Park Press, ISBN 978-0-89412-063-3. Previous publisher McGraw-Hill, New York, NY.
  5. ^ Massey, J. L. (January 1969), "Shift-register synthesis and BCH decoding" (PDF), IEEE Transactions on Information Theory, IT-15 (1): 122–127, doi:10.1109/TIT.1969.1054260, S2CID 9003708
  6. ^ Ben Atti, Nadia; Diaz-Toca, Gema M.; Lombardi, Henri (April 2006), "The Berlekamp–Massey Algorithm revisited", Applicable Algebra in Engineering, Communication and Computing, 17 (1): 75–82, arXiv:2211.11721, CiteSeerX, doi:10.1007/s00200-005-0190-z, S2CID 14944277
  7. ^ Massey 1969, p. 124

External links[edit]