Berlekamp–Massey algorithm
The Berlekamp–Massey algorithm is an algorithm that will find the shortest (LFSR) for a given binary output sequence. The algorithm will also find the of a linearly in an arbitrary . The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse. Reeds and Sloane offer an extension to handle a .
invented an algorithm for decoding Bose–Chaudhuri–Hocquenghem (BCH) codes. recognized its application to linear feedback shift registers and simplified the algorithm. Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm), but it is now known as the Berlekamp–Massey algorithm.
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Description of algorithm
The Berlekamp–Massey algorithm is an alternate method to solve the set of linear equations described in Reed–Solomon Peterson decoder, which can be summarized as:
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
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 x^{m} term shifts B(x) so it follows the syndromes corresponding to ‘b’. If the previous update of L occurred on iteration j, then m = k – j, 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 (2 L). 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.
Code sample
The algorithm from 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 + Sigma_{i=1}^L c_i * s_{n-i}; 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^{-1} x^m B(x); L = n + 1 - L; B(x) = T(x); b = d; m = 1; } else { /* step 4. */ C(x) = C(x) - d b^{-1} x^m B(x); m = m + 1; } } return L;
The algorithm for the binary field
The following is the Berlekamp–Massey algorithm specialized for the binary F_{2} (also written GF(2)). The field elements are ‘0’ and ‘1’. The field operations ‘+’ and ‘−’ are identical and are equivalent to the ‘exclusive or’ operation, XOR. The multiplication operator ‘*’ becomes the logical AND operation. The division operator reduces to the identity operation (i.e., field division is only defined for dividing by 1, and x/1 = x).
- Let be the bits of the stream.
- Initialise two arrays and each of length to be zeroes, except
- .
- For step 1 while :
- Let discrepancy be .
- if , then is already a polynomial which annihilates the portion of the stream from to .
- else:
- Let be a copy of .
- Set up to (where is the operator).
- If , set , set , and let ; otherwise leave , and alone.
At the end of the algorithm, is the length of the minimal LFSR for the stream, and we have for all .
See Also on BitcoinWiki
- , an extension for sequences over integers mod n
- Berlekamp–Welch algorithm
- , Non-Linear Feedback Shift Register