The Theory and Technology of Error Control Coding

Wireless Mobile Communication and Transmission Lab.

Chapter 2

Block code ----BCH

The Theory and Technology of Error Control Coding


Summary

The arithmetic of Galois field Linear Block Code Cyclic Code Code Construction of BCH Codes Decoding of binary BCH codes ---BMA


The arithmetic of Galois field

Self-sealed arithmetic ： If the elements of a set are still in the set by some kind of ope

ration (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).



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



Irreducible polynomial : It can not be decomposed as a product of lower degree polynomial

Example : x2 +x + 1 and x4 +x +1 are irreducible polynomials ， while x2+1 is not.

The Reason : (x+1)2= x2 +x+x + 1= x2 +1



GF(2m) There is a primitive element a GF(2^m), such that ev∈

ery 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.

i



{0 ， 1 ， a ， a2 ， a3 ， ·····,a -2 } make up the elements of GF(2m) 。

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).

m2



Minimal polynomial the minimal polynomial fi(x) of ai is the polynomial

with the smallest degree which takes ai as its root. fi

(x) has binary coefficients, and is irreducible. Conjugate Elements besides , 、、 ····· are also the elem

ents of fi(x) ， and they are called the conjugate elements of ai.

i 2i 22 i12k i


Introduction of Linear Block Code

Definition The block code whose length is n and the numbe

r of the codewords is 2k , if and only if the 2k 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.


Introduction of linear block code

Encoding:(n,k) linear block code C is a k-dimensional

subspace of Vn ， we can find k codewords which are linearly independent of each other,g0 ， g1 ， ···· ， gk-1 ， and make any codeword v in C be a linear combination of these k codewords.

v=u0g0+u1g1+····+uk-1gk-1



Taking the k-linearly independent code words as row vectors ,we can get a k*n matrix:



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.



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



Generator matrix G of systematic form can be written as follows G=[P,Ik]

The matrix G of systematic form can be got from G of non-systematic form via row/columm transformation.



Parity-check matrix For any k*n matrix G which was made up of k lin

early 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·HT=0 ,that is v·HT=0



If G is the systematic form ， then:H=[In-k,PT]



Decoding Let v= （ v0 ， v1 ， ···· ， vn-1 ） is a codeword tha

t was transmitted over a noisy channel, r= （ r0 ， r1 ， ···· ， rn-1 ） is the received code

V and R may be different,then e=v+r= （ e0 ， e1 ， ···· ， en-1 ） e is called error pattern.



Syndrome

The value of syndrome is defined as follows：



Since While v·HT=0 so we can get s=e·HT

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.


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



Generator polynomial.

Parity-check polynomial From the equality :g （ x ） ·h （ x ） =xn+1 , we can get the parity-check polynomial h （ x ）



Encoding of cyclic code Non-systematic encoding: v(x)=u(x)g(x)

Systematic encoding: v(x)=xn-ku(x)+[xn-ku(x)modg(x)]



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


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 .


BCH code

Definition: A BCH code of dmin ≥ 2td + 1 is a cyclic code whose ge

nerator polynomial g(x) has 2td consecutive roots:

So, a binary BCH (n, k, dmin) code has a generator polynomial:

121,``````,, dtbbb

g(x) = LCM{φb(x), φb+1(x), . . . , φb+2td−1(x)},Length n = LCM{nb, nb+1, . . . , nb+2td−1},and dimension k = n − deg[g(x)]


BCH code

Code Construction of BCH Codes : t-error correcting binary BCH code of length 2m 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 ro

ot 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, ，


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 sa

me minimal polynomial,

then the generator matrix can be simplified as follows:


BCH code

Example consider GF(24), p(x) = x4 + x + 1,with td = 2 and b =1. then

g(x) = LCM{φ1(x), φ3(x)} = (x4 + x + 1)(x4 + x3 + x2 + x

+ 1) = x8 + x7 + x6 + x4 + 1generates a double-error-correcting binary BCH (15,7,5)

code.


BCH code

Encoding of BCH: So we have got the generator polynomial g(x), the cod

e 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.


Decoding algorithm of BCH

BMA(Berlekamp-Massey Agorithm ） The BMA was invented by Berlekamp (1984), Masse

y (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



Let v(X)=v0+v1X+····+vn-1Xn-1 is the code polynomial of a BCH code whose length n is 2m-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)= v0+v1 i +····+vn-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



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



We assume that the transmission errors occurred,v(X)=v0+v1X+····+vn-1Xn-1 changed into

r(X)=r0+r1X+····+rn-1Xn-1 In the receiver side, r(X)=v(X)+e(x) s=(s1,s2,·····,s2t)=r·HT

so we can get the ith component of S: si= r(i)=r0+r1 i +····+rn-1 i （ n-1 ） while r(X)=ai(x)φi(X)+bi(X) φi(i)=0 si= r(i)= bi(i)=v(i)+e(i)= e(i) let



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 :



The error correcting process of BCH code ： 1 、 calculate syndrome S from the receiver polynomi

al r(X): s=(s1,s2,·····,s2t)

2 、 get the error polynomial by s1,s2,·····,s2t .

3 、 resolve the roots of ， find the position of e

rrors ,and correct the errors of r(X).

)(X

)(X


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


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:


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


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.


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:


BMA

Berlekamp–Massey algorithm:


BMA


BMA


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.

1( )


Reference

The Art of Error Correction Coding (SECOND EDITION) Written by Robert H.morelos-Zaragoza

Error Control Coding (SECOND EDITION) Written by Shu Lin Daniel J.Costello,Jr.

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The Theory and Technology of Error Control Coding

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