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.


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:

 S_{i + nu} + Lambda_1 S_{i+nu-1} + cdots + Lambda_{nu-1} S_{i+1} + Lambda_{nu} S_i = 0.

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:

 C(x) = C_{L}  x^{L} + C_{L-1}  x^{L-1} + cdots + C_2  x^2 + C_1  x + 1

or reversed:

 C(x) = 1 + C_1  x + C_2  x^2 + cdots + C_{L-1}  x^{L-1} + C_{L}  x^{L}.

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

 S_{n} + C_1  S_{n-1} + cdots + C_L  S_{n-L}

being equal to 0:

 S_{n} + C_1  S_{n-1} + cdots + C_L  S_{n-L} = 0,qquad Lle nle N-1.

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:

 d = S_{k} + C_1  S_{k-1} + cdots + C_L  S_{k-L}.

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:

C(x) = C(x)  -  (d / b)  x^m  B(x).

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 = k – j, and a recalculated discrepancy would be:

 d = S_{k} + C_1  S_{k-1} + cdots - (d/b) (S_{j} + B_1  S_{j-1} + cdots ).

This would change a recalculated discrepancy to:

 d = d - (d/b)b = d - d = 0.

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 F2 (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).

  1. Let s_0, s_1, s_2 cdots s_{n-1} be the bits of the stream.
  2. Initialise two arrays b and c each of length n to be zeroes, except b_0 leftarrow 1, c_0 leftarrow 1
  3. L leftarrow 0, m leftarrow -1.
  4. For N = 0step 1 while N < n :
    • Let discrepancy d be s_N oplus c_1s_{N-1} oplus c_2s_{N-2} oplus cdots oplus c_Ls_{N-L}.
    • if d = 0, then c is already a polynomial which annihilates the portion of the stream from N-L to N.
    • else:
      • Let t be a copy of c.
      • Set c_{N-m} leftarrow c_{N-m} oplus b_0, c_{N-m+1} leftarrow c_{N-m+1} oplus b_1, dots up to c_{n-1} leftarrow c_{n-1} oplus b_{n-N+m-1} (where oplus is the operator).
      • If L le frac{N}{2}, set L leftarrow N+1-L, set m leftarrow N, and let b leftarrow t; otherwise leave L, m and b alone.

At the end of the algorithm, L is the length of the minimal LFSR for the stream, and we have c_Ls_a oplus c_{L-1}s_{a+1} oplus c_{L-2}s_{a+2} oplus cdots = 0 for all a.

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