Wireless Information Transmission System Lab.

National Sun Yat-sen UniversityInstitute of Communications Engineering

Chapter 6

BCH Codes

2Outline

Binary Primitive BCH Codes Decoding of the BCH CodesImplementation of Galois Field ArithmeticImplementation of Error Correction Nonbinary BCH Codes and Reed-Solomon Codes

3Preface The Bose, Chaudhuri, and Hocquenghem (BCH) codes form a large class of powerful random error-correcting cyclic codes.This class of codes is a remarkable generalization of the Hamming codes for multiple-error correction.Binary BCH codes were discovered by Hocquenghem in 1959 and independently by Bose and Chaudhuri in 1960.Generalization of the binary BCH codes to codes in pm symbols (where p is a prime) was obtained by Gorenstein and Zierler.Among the nonbinary BCH codes, the most important subclass is the class of Reed-Solomon (RS) codes.Among all the decoding algorithms for BCH codes, Berlekampsiterative algorithm, and Chiens search algorithm are the most efficient ones.

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National Sun Yat-sen UniversityInstitute of Communications Engineering

Description of the Codes

5Binary Primitive BCH Codes a binary BCH codes with

Clearly, this code is capable of correcting any combination of t or fewer errors in a block of digits. We call this code a t-error-correcting BCH code. The generator polynomial of this code is specified in terms of its roots from the Galois field

. The generator poly. g(x) of the t-error-correcting BCH code of length is the lowest-degree poly. over which has

1integer 3, 2 ,mm t <

min

2 1 Block lengthNumber of parity-check digits

2 1 Minimum distance

mnn k mtd t

= +

12 = mn(2 )mGF

Let be a primitive element of (2 )mGF(2)GF

2 1m

2 3 2. . .... as its roots.t

6Binary Primitive BCH CodesIt follows from Theorem 2.11 that g(x) has and their conjugates as all its roots. Let be the minimal poly. of . Then g(x) must be the least common multiple (LCM) of

that is,

If i is an even integer, it can be expressed as a product of the following form :

where i is an odd number and Then is a conjugateof and therefore and have the same minimal poly., that is,

t22 ,...,,

{ }1 2 2( ) LCM ( ), ( ),... ( )tg x x x x =

)(xii

),(),...,(),( 221 xxx t

,2' lii =.1l lii 2)( ' =

i i 'i( ) ( ).i ix x =

7Binary Primitive BCH CodesHence, even power of has the same minimal poly. as some preceding odd power of .

The BCH codes defined above are usually called primitive (or narrow-sense ) BCH codes.

{ }1 3 2 1( ) LCM ( ), ( ),... ( )tg x x x x =

[ ]deg ( )i x m [ ] deg ( ) g x mt n k mt (Theorem 2.18 & 2.19)

8Binary Primitive BCH CodesThe single-error-correcting BCH codes of length is generated by since t =1.

Ex 6.1is a primitive element of given by Table 2.8 such

that . The minimal polynomials of ,3,5 are

is a primitive element of (2 )mGF1( ) is a primitive poly. of degree mx

0 1 22 2 2 2( , , ... , 1)m =

the single-error-correcting BCH codes of length 2 1 is a Hamming code

m

2 1m 1( ) ( )g x x=

4(2 )GF41 0 + + =

9Binary Primitive BCH Codes

The double-error-correcting BCH code of length is generated bySince and are two distinct irreducible polynomials,

)}(),({LCM)( 31 xxxg =

41

2 3 43

55

( ) 1 ( ) 1

( ) 1

x x xx x x x xx x x

= + += + + + += + +

15124 ==n

8764

4324

1 )1)(1()(

xxxxxxxxxxxg

++++=++++++=

)( )( 31 xx

min

min

(15,7) cyclic code with 5 ( ( )) 5 5

dW g x d

= =

10

Binary Primitive BCH CodesThe triple-error-correcting BCH code of length 15 is generated by

Its a (15,5) cyclic code with

v(x) is a code poly. because

{ }1 3 54 2 3 4 2

2 4 5 8 10

( ) LCM ( ), ( ), ( )

(1 )(1 )(1 ) 1

g x x x x

x x x x x x x xx x x x x x

== + + + + + + + += + + + + + +

.7,7))(( Since .7 minmin == dxgWd1

0 1 1Let ( ) ... be a poly.n

nv x v v x v x

= + + + with (2)iv GF2 2

1 2 2

If v( ) has roots , ,..., , then v( ) is divisible by ( ), ( ),..., ( ). (Theorem 2.14)

t

t

x xx x x

})(),...,(),(LCM{)(g|)(v 221 xxxxx t=

11

Binary Primitive BCH CodesWe have a new definition for t-error-correcting BCH code:A binary n-tuple v is a code word if and only if the poly. i.e.

),...,,,( 1210 = nvvvvroots as ,...,, has ...)(v 221110

tnn xvxvvx +++=

inn

iii vvvv )1(12

210 ...)(v

++++=

.21for 0

.

.

.

1

),...,,,(

)1(

1210 tivvvv

in

i

n =

12

Binary Primitive BCH CodesLet

If v is a code word in the t-error-correcting BCH code, then

The code is the null space of the matrix H and H is the parity-check matrix of the code.

=

122

122

1

)(...)(1...

.

.

.)(...)(1

...1

H

ntt

n

n

0Hv = T

13

Binary Primitive BCH Codes(Thm. 2.11)

j-th row of H can be omitted. As a result H can be reduced to the following form :

EX 6.2double-error-correcting BCH code of length (15,7) code. Let be a primitive element in

2 1

3 3 2 3 1

(2 1) 2 1 2 2 1 1

11 ( ) ( )

1 ( ) ( )

n

n

t t t n

H

=

""

"

0) v(iff 0)then v(, of conjugate a is == ijij

,15124 ==n 4(2 )GF

Row

14

Binary Primitive BCH CodesThe parity-check matrix is

Using , and representing each entry of H by its 4-tuple,

= 4263142

... 1 ... 1

H

115 =1 0 0 0 1 0 0 1 1 0 1 0 1 1 10 1 0 0 1 1 0 1 0 1 1 1 1 0 00 0 1 0 0 1 1 0 1 0 1 1 1 1 00 0 0 1 0 0 1 1 0 1 0 1 1 1 11 0 0 0 1 1 0 0 0 1 1 0 0 0 10 0 0 1 1 0 0 0 1 1 0 0 0 1 10 0 1 0 1 0 0 1 0 1 0 0 1 0 10 1 1 1 1 0 1 1 1 1 0 1 1 1 1

H

=

15

Binary Primitive BCH CodesFACT: The t-error-correcting BCH code indeed has (pf): Suppose such that

min 2 1d t + 0v

1 2Let , ,... be the nonzero components of j j jv v v v1 2 1 2 1

2 2 2 2 2

1 2

2 2

( ) ( )( ) ( )

0 ( .. )

( ) ( )

j j t j

j j t jT

j j j

j j t j

vH v v v

= =

""

# #"

1 1 2 1 2

2 2 2 2 2

2 2

( ) ( )( ) ( )

(1,1...,1) 0........

( ) ( )

j j j t

j j j t

j j j t

=

""

# # #"

(i.e. all ones)t2v)(W =

16

Binary Primitive BCH Codes1 1 2 1

2 2 2 2

2

( ) ( )( ) ( )

(1,1...,1) 0

( ) ( )

j j j

j j j

j j j

=

""

# # #"

( ), a square matrix 2A t 0A =

1 2

1 ( 1) 1

2 ( 1) 2( ... )

( 1)

11

0.......

1

j j

j jj j j

j j

+ + +

= ""

# # #"

Taking out the commonfactor from each row ofthe determinant.

17

Binary Primitive BCH CodesThe determinant in the equality above is a Vandermondedeterminant which is nonzero. The product on the left-hand side of can not be zero. This is a contradiction and hence ourassumption that there exists a nonzero code vector v of

is invalid.2t+1 is the designed distance of the t-error-correcting BCH code.The true minimum distance of a BCH code may or may not be equal to its designed distance.Binary BCH code with length can be constructed in the same manner as for the case . Let be an element of order n in , and g(x) be the binary polynomial of minimum degree that has as roots.

min 2 1d t +t2v)(W =

2 1mn (2 ) | 2 1m mGF n

12 = mnt22 ,...,,

18

Binary Primitive BCH Codesbe the minimal poly. of

respectively then

We see that g(x) is a factor of +1. The cyclic code generated by g(x) is a t-error-correcting BCH code of length n. The number of parity-check

.

If is not a primitive element of GF(2m), the code is called a nonprimitive BCH code.

2 2, ,... t

2 21, , ,..., are roots of 1

( ) ( 1)

n t n

n

x

g x x

= + +

)(),...,(),(Let 221 xxx t })(),...,(),(LCM{g(x) 221 xxx t=

nX

digits mtmin 2 1d t +

19

General definition of binary BCH codes

General definition of binary BCH codes.and consider

. , let be the minimal poly. and order of , respectively.

and the length of the code is

Note that:

or fewer errors

0(2 ), be any nonnegative integermGF A

0For 0 1i d 0 i +A

21 0000 ,...,, ++ dlll ii nx),(

{ }00 1 2LCM , ,... dn n n n =00 1 2

g( ) LCM{ ( ), ( ), ..., ( )}dx x x x =

min 0d d0parity-check digits ( 1)m d

0is capable of correcting [( 1) / 2]d

20

If we let l0 = 1, d0 = 2t+1 and be a primitive element of , the code becomes a t-error-correcting primitive BCH code of lengthIf we let l0 = 1, d0 = 2t+1 and be not a primitive element of

, the code is a nonprimitive t-error-correcting BCH code of length n, which is the order of .For a BCH code with designed distance d0, we require g(x) has d0-1 consecutive powers of a field element as roots. This guarantees that the code has . This lower bound on the minimum distance is called the BCH bound.In the rest of this chapter, we consider only the primitive BCH codes.

)GF(2m

.12 m

)GF(2m

min 0d d

General definition of binary BCH codes

Wireless Information Transmission System Lab.

National Sun Yat-sen UniversityInstitute of Communications Engineering

Decoding of the BCH Codes

22

Decoding of the BCH CodesSuppose that a code word v(x) = is transmitted and the transmission errors result :

r(x) =Let e(x) be the error pattern. Then

For decoding, remember

11

2210 ...

++++ nn xvxvxvv

11

2210 ...

++++ nn xrxrxrr

e(x)v(x)r(x) +=

=

122

122

1

)(...)(1...

.

.

.)(...)(1

...1

H

ntt

n

n

23

Decoding of the BCH CodesThe syndrome is 2t-tuple,

for

T221 Hr),...,(S == tSSS

10 1 1Let ( ) ... ( )

i i i ni ns r r r r = = + + + 1 2i t

( )can be evaluated by ( ) [ ( )]

( ) is the minimal poly. of ( ) ( ) ( ) ( )

( ) ( )

ii i x

ii

i i ii i

i i

s b x R r x

xr x a x x b xs r b

=

= + = =

24

Decoding of the BCH Codes

EX6.4Consider the double-error-correcting (15, 7) BCH code given in (from Ex6.1) . 8If (100000001000000) ( ) 1r r x x= = +

1

3

41 2 4

2 3 43

21 ( )

33 ( )

( ) ( ) ( ) 1 ( ) 1

( ) [ ( )]

( ) [ ( )] 1x

x

x x x x xx x x x x

b x R r x x

b x R r x x

= = = + += + + + += == = +

2 2 41 1 2 13 9

3 33 7

4 84 1

( ) , ( ) ( ) 1

1 ( )

s b s bs b

s b

= = = == = +

= + + == =

2 4 7 8( , , , )s =

25

Decoding of the BCH Codes

Suppose

WhereAny method for solving these equations is a decoding algorithmfor the BCH codes.

( ) 0 for 1 2 ( ) ( )i i iiv i t s r e = = =1 2

1 2( ) ... 0 ...vjj j

ve x x x x j j j n= + + +