1. Wireless Mobile Communication and Transmission Lab. Chapter 2 Block code ----BCH The Theory and Technology of Error Control Coding

2. Wireless Mobile Communication and Transmission Lab. Summary  The arithmetic of Galois field  Linear Block Code  Cyclic Code  Code Construction of BCH Codes  Decoding of binary BCH codes ---BMA

3. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  Self-sealed arithmetic ： If the elements of a set are still in the set by some kind of operation (e.g. addition,subtraction,multiplication and division),then the set is called arithmetic seal up.  Field ： the collection of arithmetic seal up elements is called field.  Finite field ： a field with finite number of elements is called a finite field or Galois field ， written as GF(2).

4. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  Binary Field A finite field which contains only the elements of 0 and 1 is called a binary field.  The arithmetic of addition and multiplication addition multiplication 0+0 ＝ 0 0*0 ＝ 0 0+1 ＝ 1 0*1 ＝ 0 1+0 ＝ 1 1*0 ＝ 0 1+1 ＝ 0 1*1 ＝ 1

5. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  Irreducible polynomial : It can not be decomposed as a product of lower degree polynomial  Example : x 2 +x + 1 and x 4 +x +1 are irreducible polynomials ， while x 2 +1 is not.  The Reason : (x+1) 2 = x 2 +x+x + 1= x 2 +1

6. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  GF(2 m ) There is a primitive element a ∈ GF(2^m), such that every nonzero element β in GF(2^m) can be expressed as, 0 ≤i ≤ 2^m − 2.  This element a is the root of an irreducible polynomial, called a primitive polynomial, p(x) over {0, 1}, that is, p(a) = 0.  A primitive element a of the field GF(2^m) satisfies the equation a^(2^m−1) = 1, and n = 2^m − 1 is the smallest positive integer such that a^n = 1.

7. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  {0 ， 1 ， a ， a 2 ， a 3 ， ·····,a -2 } make up the elements of GF(2 m ) 。  EXAMPLE: Let ‘a’ be a primitive element of GF(2^3) such that p(a) = a^3 + a + 1 = 0 and a^7 = 1. Show the ways to express the elements in GF(2^3).

8. Wireless Mobile Communication and Transmission Lab. The arithmetic of Galois field  Minimal polynomial the minimal polynomial f i (x) of a i is the polynomial with the smallest degree which takes a i as its root. f i (x) has binary coefficients, and is irreducible.  Conjugate Elements besides, 、 、 ····· are also the elements of f i (x) ， and they are called the conjugate elements of a i.

9. Wireless Mobile Communication and Transmission Lab. Introduction of Linear Block Code  Definition The block code whose length is n and the number of the codewords is 2 k, if and only if the 2 k codewords constitute a k-dimensional vector subspace of all n-dimensional vector space of the field GF (2) ， the code is called a (n, k) linear code.  Necessary and sufficient condition ： adding any two codewords using mold 2 addition, the result is still the codeword of the block code.

10. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Encoding:  (n,k) linear block code C is a k-dimensional subspace of V n ， we can find k codewords which are linearly independent of each other,g 0 ， g 1 ， ···· ， g k-1 ， and make any codeword v in C be a linear combination of these k codewords.  v=u 0 g 0 +u 1 g 1 +····+u k-1 g k-1

11. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Taking the k-linearly independent code words as row vectors,we can get a k*n matrix:

12. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  IF u= （ u0 ， u1 ， ·····uk-1 ） is the message sequence which is to be encoded, then the codeword V is as follows described: v=u·G  The linear block word C is generated by the rows of the matrix G ， so the matrix G is called the generator matrix of C.

13. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  For the linear block, the code word is divided into two parts, the message part and the redundant checking part. the message part is made up of the original message without changing, and the redundant checking part is made up of n-k parity codes. Linear block code with this construction is called systematic linear block code. Redundant checking part Message part n-k K

14. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Generator matrix G of systematic form can be written as follows G=[P,I k ]  The matrix G of systematic form can be got from G of non-systematic form via row/columm transformation.

15. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Parity-check matrix  For any k*n matrix G which was made up of k linearly independent row vectors, there exist a (n- k)*n matrix H which was made up of n-k linearly dependent row vector ， and the row vectors of H are orthogonal with any row vector of G. H is called the parity-check matrix.Thus, G·H T =0,that is v·H T =0

16. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  If G is the systematic form ， then:  H=[I n-k,P T ]

17. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Decoding  Let v= （ v 0 ， v 1 ， ···· ， v n-1 ） is a codeword that was transmitted over a noisy channel, r= （ r 0 ， r 1 ， ···· ， r n-1 ） is the received code V and R may be different, then e=v+r= （ e 0 ， e 1 ， ···· ， e n-1 ） e is called error pattern.

18. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Syndrome  The value of syndrome is defined as follows ：

19. Wireless Mobile Communication and Transmission Lab. Introduction of linear block code  Since  While v·H T =0 so we can get s=e·H T  That is  The n - k linear equations above do not have an unique solution but have 2^k solutions. To minimize the probability of a decoding error, the most probable error pattern that satisfies the equations is chosen as the true error vector.  If the channel is a BSC, the most probable error pattern is the one that has the smallest number of nonzero digits.

20. Wireless Mobile Communication and Transmission Lab. Introduction of Cyclic Code  Definition  A linear code that satisfies that any cyclic shift of a codeword is still a codeword of the linear code is called Cyclic code  Advantages: a. simpler encoded mechanism because the process can be implemented by Shift registers b. simpler decoded mechanism because of fixed algebra structure

21. Wireless Mobile Communication and Transmission Lab. Introduction of Cyclic Code  Generator polynomial.  Parity-check polynomial From the equality :g （ x ） ·h （ x ） =x n +1, we can get the parity-check polynomial h （ x ）

22. Wireless Mobile Communication and Transmission Lab. Introduction of Cyclic Code  Encoding of cyclic code Non-systematic encoding: v(x)=u(x)g(x) Systematic encoding: v(x)=x n-k u(x)+[x n-k u(x)modg(x)]

23. Wireless Mobile Communication and Transmission Lab. Introduction of Cyclic Code  General decoding of cyclic codes Let r(x) = v(x) + e(x) e(x) the error polynomial v(x) the code polynomial r(x) the received code polynomial  Then define the syndrome polynomial as

24. Wireless Mobile Communication and Transmission Lab. BCH code  BCH code which was named after the inventors (Bose- Chaudhurl-Hocquenghem), is one of cyclic code that can correct more than one errors since 1959. As the code generator polynomial is related to the minimum distance, the code construction can be easily determined by the error correcting ability. Therefore, BCH is one of the widely used error control codes.

25. Wireless Mobile Communication and Transmission Lab. BCH code  Definition: A BCH code of dmin ≥ 2t d + 1 is a cyclic code whose generator polynomial g(x) has 2td consecutive roots: So, a binary BCH (n, k, d min ) code has a generator polynomial: g(x) = LCM{φ b (x), φ b+1 (x),..., φ b+2td−1 (x)}, Length n = LCM{n b, n b+1,..., n b+2td−1 }, and dimension k = n − deg[g(x)]

26. Wireless Mobile Communication and Transmission Lab. BCH code Code Construction of BCH Codes : t-error correcting binary BCH code of length 2 m  1 Find a degree m primitive polynomial p(x) over GF(2). Use p(x) to construct GF(2^m). Let  be a primitive element of GF(2^m), i.e.  is a root of p(x) in GF(2^m). Find the Φ(x),the minimal polynomial of  ^i, for i = 1, 3,.., 2t  1. Let g(x) be the minimal degree polynomial with the Φ(x) as factors, ，

27. Wireless Mobile Communication and Transmission Lab. BCH code Simplification of generator matrix:  According to the arithmetic of finite field,  If  then, while i’ is odd.   i and  i’ are conjugate elements,so they have the same minimal polynomial, then the generator matrix can be simplified as follows:

28. Wireless Mobile Communication and Transmission Lab. BCH code  Example consider GF(2 4 ), p(x) = x 4 + x + 1,with t d = 2 and b =1. then g(x) = LCM{φ 1 (x), φ 3 (x)} = (x 4 + x + 1)(x 4 + x 3 + x 2 + x + 1) = x 8 + x 7 + x 6 + x 4 + 1 generates a double-error-correcting binary BCH (15,7,5) code.

29. Wireless Mobile Communication and Transmission Lab. BCH code  Encoding of BCH: So we have got the generator polynomial g(x), the code word v(x) can be consturcted by v(x) = b(x) + X^(n-k)*m(x). Just the same as encoding the cyclic code mentioned before.

30. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  BMA(Berlekamp-Massey Agorithm ） The BMA was invented by Berlekamp (1984), Massey (1969). This is a computationally efficient method to solve the key equation, in terms of the number of operations in GF(2^m).  EA(Euclidean Agorithm)  Direct solution

31. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  Let v(X)=v 0 +v 1 X+····+v n-1 X n-1 is the code polynomial of a BCH code whose length n is 2 m -1 and the error correcting ability is t, FOR 1<=i<=2t ，  i is the roots of v(X) ， so we can get : v(  i )= v 0 +v 1  i +····+v n-1  i(n-1) =0 From the above property,we can construct H as follows: H= 1   2  3 ··································  n-1 1  2 ( 2 ) 2 ( 2 ) 3 ···················· （  2 ） n-1 1  2t ( 2t ) 2 ( 2t ) 3 ·············· （  2t ） n-1

32. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  According to the concept of conjugate element, H can be simplified as follows:  H= 1   2  3 ··································  n-1 1  3 ( 3 ) 2 ( 3 ) 3 ···················· （  3 ） n-1 1  2t-1 ( 2t-1 ) 2 ( 2t-1 ) 3 ····· （  2t-1 ） n-1

33. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  We assume that the transmission errors occurred, v(X)=v 0 +v 1 X+····+v n-1 X n-1 changed into r(X)=r 0 +r 1 X+····+r n-1 X n-1 In the receiver side, r(X)=v(X)+e(x) s=(s 1,s 2,·····,s 2t )=r·H T so we can get the ith component of S: s i = r(  i )=r 0 +r 1  i +····+r n-1  i （ n-1 ） while r(X)=a i (x)φ i (X)+b i (X) φ i (  i )=0 s i = r(  i )= b i (  i )=v(  i )+e(  i )= e(  i ) let

34. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  From the above analysis,we can get a series of equations about the syndrome: Any method for solving these equations is a kind of decoding algorithm for BCH define the error position polynomial as follows :

35. Wireless Mobile Communication and Transmission Lab. Decoding algorithm of BCH  The error correcting process of BCH code ： 1 、 calculate syndrome S from the receiver polynomial r(X): s=(s 1,s 2, ·····,s 2t ) 2 、 get the error polynomial by s 1,s 2, ·····,s 2t. 3 、 resolve the roots of ， find the position of errors,and correct the errors of r(X).

36. Wireless Mobile Communication and Transmission Lab. BMA  The goal of the BMA is to find a (connection) polynomial σ^ (i+1)(x) of minimal degree that satisfies the following equations, derived from  (a)  The discrepancy at iteration i defined as If di = 0, then the equations (a) are satisfied for If di! = 0: Let σ ^(m)(x) be the solution at iteration m, such that −1 ≤ m < i, dm = 0,and (m − lm) is maximal. Then

37. Wireless Mobile Communication and Transmission Lab. BMA With an initial value of i = 0, the computation of σ (i+1)(x) continues until the conditions i ≥ i+1 + td − 1 or i = 2td − 1 or both are satisfied. The initial conditions of the algorithm are:

38. Wireless Mobile Communication and Transmission Lab. BMA  Example 3.5.1 Let C be the triple-error-correcting BCH (15,5,7) code of Example 3.3.3. As a reference, to check the numerical computations, the power and vector representations of GF(2^4), with primitive polynomial p(x) = 1 + x + x4, are listed. A generator polynomial for C is

39. Wireless Mobile Communication and Transmission Lab. BMA Suppose that the information polynomial is Then the corresponding code polynomial is given by Let be the polynomial associated with a vector r = v + e received after transmission of code word v over a BSC channel.

40. Wireless Mobile Communication and Transmission Lab. BMA The vector e corresponds to the error polynomial e(x) = 1 + x6 + x12. Obviously, the decoder does not know this. However, in computing the syndromes below, this knowledge is used to simplify expressions. The syndromes are:

41. Wireless Mobile Communication and Transmission Lab. BMA Berlekamp–Massey algorithm:

42. Wireless Mobile Communication and Transmission Lab. BMA

43. Wireless Mobile Communication and Transmission Lab. BMA

44. Wireless Mobile Communication and Transmission Lab. BMA Chien search and error correcting  To find the roots of σ(x), a simple trial-and-error procedure – called Chien search – is performed. All nonzero elements β of GF(2m) are generated in sequence 1, a, a^2,... And the condition = 0 tested.  For binary BCH codes, once the error locations j1,..., jv are known, the corresponding bits in the received word are complemented and the estimated code word v(x) generated.

45. Wireless Mobile Communication and Transmission Lab. Reference  The Art of Error Correction Coding (SECOND EDITION) Written byRobert H.morelos-Zaragoza  Error Control Coding (SECOND EDITION) Written by Shu Lin Daniel J.Costello,Jr.