1. Berlekamp Massey: Euclid in Disguise
Ishai Ilani
March 2018
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11/24/2018

2. Outline BCH Code – Parity Check Matrix Key Equation Derivation
Euclid and BM Algorithms
Definition of Backward Polynomial Division
Definition of Backward Euclid Algorithm
BM Algorithm as a Backward Euclid Algorithm
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11/24/2018

3. BCH Parity Check matrix
For a primitive Galois field element 𝛼, and a matrix 𝐻 given by:
𝐻= 1 𝛼 𝛼 2 ⋯ ⋯ 𝛼 𝑛−1 1 𝛼 2 𝛼 2∙2 ⋯ ⋯ 𝛼 𝑛−1 ∙2 1 𝛼 3 𝛼 3∙2 ⋯ ⋯ 𝛼 𝑛−1 ∙3 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1 𝛼 2𝑡 𝛼 2𝑡∙2 ⋯ ⋯ 𝛼 𝑛−1 ∙2𝑡
the code 𝐶= 𝒄∈𝐺𝐹 2 𝑛 | 𝐻𝒄=0 is a BCH code.
If and 𝒄+𝒆 is read, (𝒆 - an error vector), then 𝐻 𝒄+𝒆 =𝐻𝒆.
Denote the support of 𝒆 as: 𝑒 𝑖 1 , 𝑒 𝑖 2 ,…, 𝑒 𝑖 𝐿 .
The vector 𝐻𝒆, also known as the syndrome vector, is given by:
𝑆 1 𝑆 2 𝑆 3 ⋮ 𝑆 2𝑡 = 𝛼 𝑖(1) 𝛼 𝑖(1)∙2 𝛼 𝑖(1)∙3 ⋮ 𝛼 𝑖(1)∙2𝑡 𝛼 𝑖(2) 𝛼 𝑖(2)∙2 𝛼 𝑖(2)∙3 ⋮ 𝛼 𝑖(2)∙2𝑡 +⋯+ 𝛼 𝑖(𝐿) 𝛼 𝑖(𝐿)∙2 𝛼 𝑖(𝐿)∙3 ⋮ 𝛼 𝑖(𝐿)∙2𝑡
Geometric Progressions
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11/24/2018

4. Errors and Geometric Progressions
Each column of 𝐻 is a geometric progression.
The syndrome vector 𝑺 is a sum of 𝐿 geometric progressions, (𝐿 – the error vector support).
CONCLUSION:
Finding the errors of a BCH code is equivalent to finding a minimal number of geometric progressions such that their sum is equal to the syndrome vector.
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11/24/2018

5. Error Locator Polynomial (ELP)
Identify vectors with polynomials
𝑠 1 , 𝑠 2 ,⋯, 𝑠 2𝑡 𝑇 ↔𝑆 𝐷 = 𝑗=1 2𝑡 𝑠 𝑗 𝐷 𝑗−1
𝑆 𝐷 = 𝑙=1 𝐿 𝑌 𝑖 𝑙 𝛼 𝑖(𝑙) 𝑗=0 2𝑡−1 𝛼 𝑖 𝑙 𝐷 𝑗 = 𝑙=1 𝐿 𝑌 𝑖 𝑙 𝛼 𝑖(𝑙) 1− 𝛼 𝑖(𝑙) 𝐷 2𝑡 1− 𝛼 𝑖(𝑙) 𝐷
ELP = 𝐶 𝐷 = 𝑙=1 𝐿 1− 𝛼 𝑖(𝑙) 𝐷 =𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
NOTE: For each 𝑙, 𝐶(𝐷) 1− 𝛼 𝑖 𝑙 𝐷 is a polynomial of order L−1.
𝑆 𝐷 𝐶 𝐷 = 𝑙=1 𝐿 𝑌 𝑖 𝑙 𝛼 𝑖(𝑙) ∙𝐶(𝐷) 1− 𝛼 𝑖 𝑙 𝐷 𝑚𝑜𝑑 𝐷 2𝑡
For 𝐿≤𝑖<2𝑡 the coefficients of 𝐷 𝑖 in 𝑆 𝐷 𝐶 𝐷 are 0.
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6. Key Equation Denote: 𝛺 𝐷 = 𝑙=1 𝐿 𝑌 𝑖(𝑙) 𝛼 𝑖 𝑙 1− 𝛼 𝑖 𝑙 𝐷 𝐶(𝐷) .
Note that:
deg 𝛺 𝐷 < deg 𝐶 𝐷 =𝐿=#𝐸𝑟𝑟𝑜𝑟𝑠
𝑐 0 ≠0
Key Equation: 𝑆 𝐷 𝐶 𝐷 = 𝛺 𝐷 𝑚𝑜𝑑 𝐷 2𝑡
Rephrase as: 𝑅 𝐷 =𝑆 𝐷 𝐶 𝐷 −𝛺 𝐷 =0 𝑚𝑜𝑑 𝐷 2𝑡 * 2𝑡
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7. Solving the Key Equation
Euclid Algorithm
The (ordinary) Euclid algorithm applies to the pair 𝐺(𝐷)= 𝐷 2𝑡 , 𝐻(𝐷)=𝑆(𝐷).
𝐺 0 =𝐺, 𝐻 0 =𝐻, 𝐺 𝑖 = 𝐻 𝑖−1 , 𝐻 𝑖 = 𝐺 𝑖−1 𝑚𝑜𝑑 𝐻 𝑖−1 .
It terminates at the first iteration 𝑖 for which deg⁡( 𝐺 𝑖 𝑚𝑜𝑑 𝐻 𝑖 )<𝑡.
At this point denote 𝛺= 𝐺 𝑖 𝑚𝑜𝑑 𝐻 𝑖 .
Compute the (minimal degree) Bezout pair (𝐴,𝐶), such that 𝛺=𝐴𝐺+𝐶𝐻.
𝐶 is the ELP and 𝛺 is the evaluator polynomial. 1
𝑠 1
𝑠 2
𝑠 3 ⋯
𝑠 2𝑡
𝐺 0 =𝐺
𝐻 0 =𝐻
Berlekamp Massey (BM) Algorithm
Synthesize the shortest LFSR generating the syndrome 𝑆.
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11/24/2018

8. Massey Presentation of Berlekamp Algorithm
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9. BM – Pros, Cons and Notation
Simple to implement
Cons
Not intuitive
In tribute to Massey we follow Massey’s notation.
Thus the polynomials are presented in the indeterminate 𝐷.
The parameters 𝑆, 𝐶, 𝐵 and 𝐿 have similar roles to their roles in Massey’s paper.
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11/24/2018

10. Solving the Key Equation by Backward Euclid Algorithm
Main Proposition
Solving the Key Equation by Backward Euclid Algorithm
The Key Equation may be solved by a Backward Euclid Algorithm applied to the polynomials 𝐺 𝐷 =𝐷∙𝑆 𝐷 = 𝑖=1 2𝑡 𝑠 𝑖 𝐷 𝑖 , 𝐻 𝐷 =1 1
𝑠 1
𝑠 2
𝑠 3 ⋯
𝑠 2𝑡
𝐻 0 =𝐻=1
𝐺 0 =𝐺=𝐷𝑆(𝐷)
The BM algorithm is a Backward Euclid Algorithm applied to the polynomials 𝐺 𝐷 =𝐷∙𝑆 𝐷 = 𝑖=1 2𝑡 𝑠 𝑖 𝐷 𝑖 , 𝐻 𝐷 =1+𝐺(𝐷) 1
𝑠 1
𝑠 2
𝑠 3 ⋯
𝑠 2𝑡
𝐻 0 =1+𝐺(𝐷)
𝐺 0 =𝐺=𝐷𝑆(𝐷)
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11/24/2018

11. ם ו ל ש Crash Course in Hebrew Peace degree e e r g e d modulo o l u d
For a polynomial 𝒑(𝐷)= 𝑝 𝑛 𝐷 𝑛 + 𝑝 𝑛−1 𝐷 𝑛−1 +⋯ 𝑝 𝑙+1 𝐷 𝑙+1 + 𝑝 𝑙 𝐷 𝑙 with 𝑝 𝑛 ≠0, 𝑝 𝑙 ≠0 deg 𝒑 𝐷 =𝑛 ged 𝒑 𝐷 =𝑙 In other words for any polynomial 𝒑 𝐷 𝒑 𝐷 = 𝑖=𝑔𝑒𝑑(𝒑) deg⁡(𝒑) 𝑝 𝑖 𝐷 𝑖 *
𝑔𝑒𝑑(𝒑)
𝑑𝑒𝑔(𝒑)
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12. Polynomial Division - I
Ordinary Polynomial Division
Backward Polynomial Division
𝐷 3
𝐷 4 + 𝐷 3
+ 𝐷
+ 1
𝐷+1
⋯⋯+
⋯⋯ Continue forever
𝐷 6 + 𝐷 5
𝐷 5 +
𝐷 6
𝐷 5
𝐷 4 +
𝐷 5 + 𝐷 4
𝐷 3 + 𝐷 2
𝐷 2
𝐷+1
𝐷 4 + 𝐷 3 + 𝐷 2 +0𝐷+0
𝐷 4 + 𝐷 3 + 𝐷 2 +0𝐷+0
𝐷+1
𝐷 2 +0𝐷+0
𝐷 4
𝐷 2 +𝐷
deg 𝑅 =deg⁡(𝐻)
𝐷+0 1
deg 𝑅 <deg⁡(𝐻)
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13. Bezout Pairs and Minimal Degree Bezout Pairs
A Bezout pair (𝐴 𝐷 , 𝐵(𝐷)) relative to (𝐺,𝐻) is a pair of polynomials associated with 𝐴 𝐷 𝐺 𝐷 +𝐵 𝐷 𝐻(𝐷).
Define the degree of any vector of polynomials 𝑷 𝐷 =( 𝑃 1 𝐷 , 𝑃 2 𝐷 , …,𝑃 𝑛 𝐷 ) as
deg 𝑷 𝐷 = max 𝑖 (deg⁡( 𝑃 𝑖 𝐷 ) .
It follows that deg 𝐴, 𝐵 =max⁡{ deg 𝐴 , deg⁡(𝐵)}
Example 𝐺 𝐷 = 𝐷 4 + 𝐷 3 + 𝐷 2 , 𝐻 𝐷 =1.
𝐺 may be represented by the Bezout pair 𝑏𝑝 𝐺 =(1,0), and may also be represented by the Bezout pair 𝑏𝑝 𝐺 =(0, 𝐺(𝑥)).
The first representation is a minimal degree representation.
Under certain conditions minimal degree Bezout pairs are unique, (e.g. if 𝐺,𝐻 are relatively prime and deg⁡(𝐴,𝐵) < deg⁡(𝐺,𝐻) ).
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14. Backward Polynomial Division – Stopping Condition
Let 𝐺,𝐻 be polynomials with 𝑔𝑒𝑑 𝐺 ≥𝑔𝑒𝑑(𝐻).
Perform Backward Polynomial Division on 𝐺,𝐻 and in parallel on their Bezout pairs, and stop when deg⁡(𝑏𝑝(𝑅))>deg⁡(𝑏𝑝(𝐺)).
Backward Polynomial Division
Bezout Pairs
𝑅↔ 1, 0 , 𝐺↔ 1,0 , 𝐻↔(0,1) 𝑅=
𝐷 3 + 𝐷 2
𝑄=𝐷 2
𝐺=𝐷 4 + 𝐷 3 + 𝐷 2 +0𝐷+0
𝐷+1=𝐻
𝑅=𝐷 4
𝑅↔ 1, 𝐷 2 , 𝐺↔ 1,0 , 𝐻↔(0,1)
𝑑𝑒𝑔 𝑏𝑝 𝑅 >𝑑𝑒𝑔(𝑏𝑝 𝐺 )
𝑅=𝐷 4
𝑄= 𝐷 2
𝑅=𝐺− 𝐷 2 𝐻
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15. Polynomial Division - Summary
Ordinary Polynomial Division
Backward Polynomial Division
𝐺 and 𝐻 with 𝑑𝑒𝑔 𝐺 ≥𝑑𝑒𝑔⁡(𝐻)
𝐺 and 𝐻 with 𝑔𝑒𝑑 𝐺 ≥𝑔𝑒𝑑(𝐻).
𝑅=𝐺, 𝑄=0
𝐿𝑜𝑜𝑝 𝑢𝑛𝑡𝑖𝑙 𝑑𝑒𝑔 𝑅 < 𝑑𝑒𝑔 𝐻
𝐿𝑜𝑜𝑝 𝑢𝑛𝑡𝑖𝑙 𝑑𝑒𝑔 𝑏𝑝(𝑅) > 𝑑𝑒𝑔 𝑏𝑝(𝐺)
𝑞= 𝑢 𝑅 𝑢 𝐻 𝐷 𝑑𝑒𝑔 𝑅 −𝑑𝑒𝑔(𝐻)
𝑞= 𝑙 𝑅 𝑙 𝐻 𝐷 𝑔𝑒𝑑 𝑅 −𝑔𝑒𝑑(𝐻)
𝑅=𝑅−𝑞𝐻
𝑄=𝑄+𝑞
𝑅≝𝐺 𝑚𝑜𝑑 𝐻
𝑅≝𝐺 𝑑𝑜𝑚 𝐻
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16. Euclid Algorithm and Backward Euclid Algorithm
Recall: For the Backward Polynomial Division (BPD) we have
𝑅=𝐺−𝑄𝐻, 𝑅≝𝐺 𝑑𝑜𝑚 𝐻
Define: 𝑄 2 ≝ Σ 𝑗=𝑔𝑒𝑑(𝑄 𝑑𝑒𝑔 𝑄 −1 𝑞 𝑗 𝐷 𝑗
We can now define a Backward Euclid Algorithm (BEA) which mirrors the well- known Ordinary Euclid Algorithm.
Ordinary Euclid
Backward Euclid
𝐺 0 =𝐺, 𝐻 0 =𝐻
𝐺 𝑖 = 𝐻 𝑖−1
𝐻 𝑖 = 𝐺 𝑖−1 − 𝑄 2 𝑖−1 𝐻 𝑖−1
𝐻 𝑖 = 𝐺 𝑖−1 𝑚𝑜𝑑 𝐻 𝑖−1
𝐺 𝑖 = 𝐺 𝑖−1 𝑑𝑜𝑚 𝐻 𝑖−1
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17. Berlekamp Massey as Backward Euclid
Let 𝑆 𝐷 = 𝑖=1 2𝑡 𝑠 𝑖 𝐷 𝑖−1 be the syndrome polynomial of a BCH code.
Set 𝔊 𝐷 =𝐷𝑆 𝐷 , ℌ 𝐷 =1, 𝐺 𝐷 =𝔊, 𝐻 𝐷 =𝔊+ℌ.
𝑏𝑝 𝐺 = 1,0 , 𝑏𝑝 𝐻 =(1,1).
The BM algorithm is actually a BEA applied to 𝐺 𝐷 , 𝐻 𝐷 with Bezout pairs computed relative to 𝔊, ℌ.
Precisely, the BEA is applied until we reach a 𝐺 𝑖 satisfying 𝑔𝑒𝑑( 𝐺 𝑖 )>2𝑡.
If also 𝑔𝑒𝑑( 𝐻 𝑖 )>2𝑡 set 𝑅=𝐻 𝑖 , else set 𝑅=𝐺 𝑖 .
Compute the Bezout pair of 𝑅, 𝑏𝑝 𝑅 = 𝐶,𝑊 , then:
𝐶 is the ELP, and 𝛺≝−𝑊/𝐷 is the evaluator polynomial.
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11/24/2018

18. Sanity Check 𝑅=𝐶𝔊+𝑊ℌ=0 𝑚𝑜𝑑 ( 𝐷 2𝑡+1 ) 𝑅=𝐷𝐶 𝐷 𝑆 𝐷 +𝑊 𝐷 =0 𝑚𝑜𝑑 𝐷 2𝑡+1
𝑅=𝐷𝐶 𝐷 𝑆 𝐷 +𝑊 𝐷 =0 𝑚𝑜𝑑 𝐷 2𝑡+1
Dividing by 𝐷: 𝐶 𝐷 𝑆 𝐷 +𝑊 𝐷 /𝐷=0 𝑚𝑜𝑑 𝐷 2𝑡
Or: 𝐶 𝐷 𝑆 𝐷 =𝛺(𝐷) 𝑚𝑜𝑑 𝐷 2𝑡 ,
and 𝑔𝑒𝑑 𝐶 =0
If deg 𝐶,𝑊 ≤𝑡, then deg 𝐶 ≤𝑡, deg 𝛺 <𝑡 is the unique pair with these properties, so 𝐶 𝐷 is the ELP.
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11/24/2018

19. A Taste of the Proof
Theorem 1: Let (𝐶,𝛺) be a polynomial pair with 𝑔𝑒𝑑(𝐶)=0, and
𝑔𝑒𝑑 𝐶∙𝑆−𝛺 =𝑁.
Let (𝐶2,𝛺2) be another polynomial pair with 𝑔𝑒𝑑(𝐶2)=0 satisfying
𝑔𝑒𝑑 𝐶2∙𝑆−Ω2 >𝑁.
Then: 𝑑𝑒𝑔 𝐶2,𝐷∙𝛺2 +𝑑𝑒𝑔 𝐶,𝐷∙𝛺 ≥𝑁+1.
Proof: For 𝑑𝑒𝑔 𝐶,𝐷∙𝛺 ≥𝑁+1 the statement is trivial.
So assume 𝑑𝑒𝑔 𝐶,𝐷∙𝛺 ≤𝑁, and assume 𝑑𝑒𝑔 𝐶,𝐷∙𝛺 +𝑑𝑒𝑔 𝐶2,𝐷∙𝛺2 ≤𝑁.
It is straight forward to see that
𝑔𝑒𝑑 𝐶2 𝐶∙𝑆−Ω =𝑁, but 𝑔𝑒𝑑 𝐶 𝐶2∙𝑆−Ω2 >𝑁.
Subtracting, we get: 𝑔𝑒𝑑 𝐶∙𝛺2−𝐶2∙𝛺 =𝑁.
But according to the assumptions
𝑑𝑒𝑔 𝐶∙𝛺2−𝐶2∙𝛺 ≤𝑚𝑎𝑥 𝑑𝑒𝑔 𝐶∙𝛺2 ,deg⁡ 𝐶2∙Ω <𝑁.
CONTRADICTION !
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20. Proof - II
Assume
𝑔𝑒𝑑 𝐻 𝑖 =𝑑𝑒𝑔 𝑏𝑝( 𝐺 𝑖 ) +𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖 ) 𝐿 𝑔𝑒𝑑 𝐻 𝑖 −1 𝑆 =𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖 ) 𝐿 𝑔𝑒𝑑 𝐻 𝑖 𝑆 =𝐿 𝑔𝑒𝑑 𝐺 𝑖 −1 𝑆 =𝑑𝑒𝑔 𝑏𝑝( 𝐺 𝑖 ) ⋯ ∗
𝐻 𝑖
𝐺 𝑖
𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖 )
𝑑𝑒𝑔 𝑏𝑝( 𝐺 𝑖 ) =
𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖+1 )
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11/24/2018

21. Proof Continued 𝐺 𝑖 𝐻 𝑖+1 𝐷 Δ 𝐻 𝑖 𝐺 𝑖+1 ⋯ ∗ ⋯ ∗ ⋯ ∗ ⋯ ∗ Δ Δ∗
𝐺 𝑖 ⋯ ∗
𝐻 𝑖+1 Δ ⋯ ∗
𝐷 Δ 𝐻 𝑖 Δ
𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖 )
𝑑𝑒𝑔 𝑏𝑝( 𝐻 𝑖+1 )
𝑑𝑒𝑔 𝑏𝑝( 𝐺 𝑖+1 ) ⋯ ∗
𝐺 𝑖+1
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22. References
E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, chapter 7 (1968).
J.L. Massey, “Shift register synthesis and BCH decoding”, IEEE Trans. Inform. Theory, vol. IT-15, pp. 122–127 (1969).
D. V. Sarwate and N. R. Shanbhag, "High-speed architectures for Reed-Solomon decoders," in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 9, no. 5, pp , Oct
F.J. MacWilliams,‎ N.J.A. Sloane, The Theory of Error-Correcting Codes, Volume 16, North-Holland Mathematical Library,
K. Imamura and W. Yoshida, “A simple derivation of the Berlekamp- Massey algorithm and some applications”, in IEEE Transactions on Information Theory, vol. 33, no. 1, pp , Jan 1987.
M. Bras-Amoros M. E. O’Sullivan, “The Berlekamp-Massey Algorithm and the Euclidean Algorithm: a Closer Link”, [Online.] Available:
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©2017 Western Digital Corporation or its affiliates. All rights reserved.
11/24/2018

23. ©2017 Western Digital Corporation or its affiliates
©2017 Western Digital Corporation or its affiliates. All rights reserved.
11/24/2018