The Berlekamp–Massey algorithm

The Berlekamp–Massey algorithm inverts a Toeplitz system of equations, in any field F, of the form

The Berlekamp–Massey algorithm is formally valid in any field, but it may suffer from problems of numerical precision in the real field or the complex field. The computational problem it solves is the same problem solved by the Sugiyama algorithm.

The Berlekamp–Massey algorithm can be described as a fast algorithm for finding a linear recursion, of shortest length, of the form

E_j = −
^ν

k=1

kE_{(( j−k))} j= ν, . . . , 2t − 1.

This is the shortest linear recursion that produces E_ν,. . . , E2t−1from E₀,. . . , E_ν−1. This formulation of the problem statement is actually stronger than the problem of solving the matrix equation because the matrix equation may have no solution. If the matrix equation has a solution, then a minimum-length linear recursion of this form exists, and the Berlekamp–Massey will find it. If the matrix equation has no solution, then the Berlekamp–Massey algorithm finds a linear recursion of minimum length, but with ν > t, that produces the sequence. Thus the Berlekamp–Massey algorithm actually provides more than was initially asked for. It always computes the linear complexity L=L(E0,. . . , E2t−1) and a corresponding shortest linear recursion that will produce the given sequence.

The Berlekamp–Massey algorithm, shown in Figure3.1, is an iterative procedure for finding a shortest cyclic recursion for producing the initial r terms, E₀, E₁,. . . , Er−1, of the sequence E. At the rth step, the algorithm has already computed the linear recursions(⁽ⁱ⁾(x), Li) for all i smaller than r. These are the linear recursions that, for each i, produce the first i terms of the sequence E. Thus for each i= 0, . . . , r − 1, we have already found the linear recursion

E_j = −

Li

k=1

⁽ⁱ⁾_k E_j−k j= Li,. . . , i

for each i smaller than r. Then for i= r, the algorithm finds a shortest linear recursion that produces all the terms of the sequence E. That is, it finds the linear recursion (^(r)(x), Lr) such that

E_j = −

L_r

k=1

^(r)_k E_j−k j= Lr,. . . , r.

The rth step of the algorithm begins with a shortest linear recursion, (^(r−1)(x), Lr−1), that produces the truncated sequence E^r−1 = E0, E₁,. . . , Er−1. Define

δr = Er−

⎛

⎝−^L
r−1

k=1

^(r−1)_k E_r−k

⎞

⎠

=

L
_r−1

k=0

^(r−1)_k E_r−k

157 3.5 The Berlekamp–Massey algorithm

Initialize

(x) = B(x) = 1

No ∆ ^{= 0} Yes

?

?

L = r = 0

2L ≤ r – 1

r ← r+1

r = 2t Yes No

Halt

d = 0 d = 1

L ← r – L

(x) B (x)

(x) B (x) –∆x

(1 – d)x I

∆^–1d

n –1

j = 0

Σ

∆ =
j E_{r – j}

Figure 3.1. Berlekamp–Massey algorithm.

as the discrepancy in the output of the recursion at the rth iteration. The discrepancyδr

need not be zero. Ifδris zero, the output of the recursion is the desired field element.

It is then trivial to specify the next linear recursion. It is the same linear recursion as found in the previous iteration. In this case, set

(^(r)(x), Lr) = (^(r−1)(x), Lr−1)

as a shortest linear recursion that produces the truncated sequence E^r, and the iteration is complete. In general, however,δrwill be nonzero. Then

(^(r)(x), Lr) = (^(r−1)(x), Lr−1).

To see how^(r−1)(x) must be revised to get ^(r)(x), choose an earlier iteration count m, smaller than r, such that

L
_m−1

By translating indices so that j+ m − r replaces j and then scaling, this becomes δr

where δm is nonzero and E_j is regarded as zero for j negative. This suggests the polynomial update second sum equalsδm. Therefore

L_r

and the new polynomial^(r)(x) provides a new linear recursion that produces one more symbol than the previous linear recursion.

To ensure that the recursion is a minimum-length recursion, we need to place an additional condition on the choice of^(m−1)(x). Until now, we only required that m be chosen so thatδm= 0. Now we will further require that m be the most recent index such that L_m> Lm−1. This requirement implies the earlier requirement thatδm= 0, so that condition need not be checked. The following theorem shows that this last condition ensures that the new recursion will be of minimum length. By continuing this process for 2t iterations, the desired recursion is found.

159 3.5 The Berlekamp–Massey algorithm

Theorem 3.5.1 (Berlekamp–Massey) Suppose thatL(E0, E₁,. . . , Er−2) = L. If the recursion ((x), L) produces E0, E₁,. . . , Er−2, but ((x), L) does not produce E0, E₁,. . . , Er−1, thenL(E0, E₁,. . . , Er−1) = max[L, r − L].

Proof: Let E^(r)= E0, E₁,. . . , Er−1. Massey’s theorem states that L(E^(r)) ≥ max[L, r − L].

Thus it suffices to prove that L(E^(r)) ≤ max[L, r − L].

Case (1) E^(r) = (0, 0, . . . , 0, Er−1 = 0). The theorem is immediate in this case because a linear shift register of length zero produces E^(r−1)= (0, 0, . . . , 0), while a linear shift register of length r is needed to produce E^(r) = (0, 0, . . . , 0, Er−1).

Case (2) E^(r−1) = (0, 0, . . . , 0). The proof is by induction. Let m be such that L(E^(m−1)) < L(E^(m)) = L(E^(r−1)). The induction hypothesis is that L(E^(m)) = max[Lm−1, m− Lm−1]. By the construction described prior to the theorem,

L(E^(r)) ≤ max[L, Lm−1+ r − m].

Consequently,

L(E^(r)) ≤ max[L(E^(r−1)), r −L(E^(r−1))]

= max[L, r − L],

which proves the theorem.

Corollary 3.5.2 (Berlekamp–Massey algorithm) In any field, let S₁,. . . , S2t be given. Under the initial conditions⁽⁰⁾(x) = 1, B⁽⁰⁾(x) = 1, and L0 = 0, let the following set of equations be used iteratively to compute^(2t)(x):

δr=

L
_r−1

j=0

^(r−1)_j S_r−j,

L_r= r(r − Lr−1) + (1 − r)Lr−1,

^(r)(x) B^(r)(x)



=



1 −δrx δ⁻¹r r (1 − r)x

 ^(r−1)(x) B^(r−1)(x)

 ,

r = 1, . . . , 2t, where r= 1 if both δr= 0 and 2Lr−1 ≤ r − 1, and otherwise r= 0.

Then^(2t)(x) is the polynomial of smallest degree with the properties that ^(2t)₀ = 1,

and

S_r+

L_r−1

j=1

^(2t)_j S_r−j= 0 r = L2t,. . . , 2t − 1.

The compact matrix formulation given in the corollary includes the term δ_r⁻¹r. Becauseδrcan be zero only whenris zero, the termδ_r⁻¹ris then understood to be zero. The Berlekamp–Massey algorithm, as shown in Figure3.1, saves the polynomial

(x) whenever there is a length change as the “interior polynomial” B(x). This B(x) will play the role of^(m−1)(x) when it is needed in a later iteration. In Corollary3.5.2, the interior polynomial B(x) is equal to δ_m⁻¹x^r−m^(m)(x). When r = 1, B(x) is replaced by(x), appropriately scaled, and when r= 0 it is multiplied by x to account for the increase in r.

Note that the matrix update requires at most 2t multiplications per iteration, and the calculation ofδr requires no more than t multiplications per iteration. There are 2t iterations and hence at most 6t² multiplications. Thus using the algorithm will usually be much better than using a matrix inversion, which requires on the order of t³ multiplications.

The Berlekamp–Massey algorithm is formally valid in any field. However, the deci-sion to branch is based on whether or notδrequals zero, so in the real field the algorithm is sensitive to problems of computational precision.

A simple example of the iterations of the Berlekamp–Massey algorithm in the rational field is shown in Table 3.2. In this example, the algorithm computes the shortest recursion that will produce the sequence 1, 1, 0, 1, 0, 0 in the rational field.

A second example of the iterations of the Berlekamp–Massey algorithm in the field GF(16) is shown in Table3.3. In this example, the algorithm computes the shortest recursion that will compute the sequenceα¹², 1,α¹⁴,α¹³, 1,α¹¹ in the field GF(16).

This is the sequence of syndromes for the example of the (15, 9, 7) Reed–Solomon code, using the same error pattern that was studied earlier in Section3.1. As before, the senseword is the all-zero codeword, andα is the primitive element of GF(16) that satisfiesα⁴= α + 1.

Now we turn to the final task of this section, which is to exploit the structure of the Berlekamp–Massey algorithm to improve the Forney formula by eliminating the need to compute(x).

Corollary 3.5.3 (Horiguchi–Koetter) Suppose(x) has degree ν. The components of the error vector e satisfy

e_i=

⎧⎨

⎩

0 if(ω⁻ⁱ) = 0 ω^−i(ν−1)

ω⁻ⁱB(ω⁻ⁱ)^(ω⁻ⁱ) if(ω⁻ⁱ) = 0,

161 3.5 The Berlekamp–Massey algorithm

Table 3.2. Example of Berlekamp–Massey algorithm for a sequence of rationals S₀= 1

S₁= 1 S₂= 0 S₃= 1 S₄= 0 S₅= 0

r δr B(x) (x) L

0 1 1 0

1 1 1 1− x 1

2 0 x 1− x 1

3 −1 −1 + x 1− x + x² 2

4 2 −x + x² 1+ x − x² 2

5 1 1+ x − x² 1+ x − x³ 3

6 0 x+ x²− x³ 1+ x − x³ 3

(x) = 1 + x − x³

Table 3.3. Example of Berlekamp–Massey algorithm for a Reed–Solomon (15, 9, 7) code

g(x) = x⁶+ α¹⁰x⁵+ α¹⁴x⁴+ α⁴x³+ α⁶x²+ α⁹x+ α⁶ v(x) = αx⁷+ α⁵x⁵+ α¹¹x² = e(x)

S₁= αα⁷+ α⁵α⁵+ α¹¹α²= α¹² S₂= αα¹⁴+ α⁵α¹⁰+ α¹¹α⁴= 1 S₃= αα²¹+ α⁵α¹⁵+ α¹¹α⁶= α¹⁴ S₄= αα²⁸+ α⁵α²⁰+ α¹¹α⁸= α¹³ S₅= αα³⁵+ α⁵α²⁵+ α¹¹α¹⁰= 1 S₆= αα⁴²+ α⁵α³⁰+ α¹¹α¹²= α¹¹

r δr B(x) (x) L

0 1 1 0

1 α¹² α³ 1+ α¹²x 1

2 α⁷ α³x 1+ α³x 1

3 1 1+ α³x 1+ α³x+ α³x² 2

4 1 x+ α³x² 1+ α¹⁴x 2

5 α¹¹ α⁴+ α³x 1+ α¹⁴x+ α¹¹x²+ α¹⁴x³ 3 6 0 α⁴x+ α³x² 1+ α¹⁴x+ α¹¹x²+ α¹⁴x³ 3

(x) = 1 + α¹⁴x+ α¹¹x²+ α¹⁴x³

= (1 + α⁷x)(1 + α⁵x)(1 + α²x)

where B(x) is the interior polynomial computed by the Berlekamp–Massey algorithm.

Proof: The actual number of errors isν, the degree of (x). Define the modified error vector˜e by the components ˜ei = eiB(ω⁻ⁱ). To prove the corollary, we will first show that B(ω⁻ⁱ) is nonzero everywhere that ei is nonzero. Then we will apply the Forney formula to the modified error vector˜e, and finally divide out B(ω⁻ⁱ).

The iteration equation of the Berlekamp–Massey algorithm can be inverted as follows:

(1 − r)x δrx

−δ⁻¹r r 1

 ^(r)(x) B^(r)(x)



= x

^(r−1)(x) B^(r−1)(x)

 .

If^(r)(x) and B^(r)(x) have a common factor other than x, then ^(r−1)(x) and B^(r−1)(x) have that factor also. Hence by induction,⁽⁰⁾(x) and B⁽⁰⁾(x) also have that same factor.

Because^(r)(x) does not have x as a factor, and because ⁽⁰⁾(x) = B⁽⁰⁾(x) = 1, there is no common factor. Therefore

GCD[(x), B(x)] = 1.

Because (x) and B(x) are coprime, they can have no common zero. This means that the modified error component˜ei is nonzero if and only if error component e_i is nonzero. Consequently,(x) is also the error-locator polynomial for the modified error vector˜e. For the modified error vector, the syndromes are

S_j=

n−1

i=0

e_iB(ω⁻ⁱ)ω^ij j= 0, . . . , 2t − 1

=

n−1

k=0

B_kS_j−k

=



0 j< ν − 1 1 j= ν − 1

where the second line is a consequence of the convolution theorem, and the third line is a consequence of the structure of the Berlekamp–Massey algorithm. Thus S(x) = x^ν−1. The modified error-evaluator polynomial for the modified error vector is given by

(x) = (x)S(x) (mod x^ν)

= x^ν−1.

163 3.6 Decoding of binary BCH codes

The Forney algorithm, now applied to the modified error vector, yields

˜ei = − (ω⁻ⁱ) ω⁻ⁱ^(ω⁻ⁱ),

from which the conclusion of the corollary follows.

In document Algebraic Codes on Lines, Planes, And Curves (Page 179-187)