14.Lecture

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Lecture 14Lecture 14

BCH CodesBCH Codes

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99 A class of cyclic codes discoveredA class of cyclic codes discovered

independently byindependently byHocquenghemHocquenghem in 1959 andin 1959 andBose & RayBose & Ray--ChaudhuriChaudhuri in 1960.in 1960.

99

One of the most powerful known classes ofOne of the most powerful known classes ofLinear Cyclic Block CodesLinear Cyclic Block Codes

99 Capable of correcting multiple errorsCapable of correcting multiple errors

99 Easy Encoding and DecodingEasy Encoding and Decoding

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FeaturesFeatures

AA BCHBCH Code can be defined for anyCode can be defined for any

positive integerspositive integers mm ( m 3) and any( m 3) and any tt(t

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Generator PolynomialGenerator Polynomial

This code is capable of correcting anyThis code is capable of correcting any

combination ofcombination oftt or fewer errors in a block lengthor fewer errors in a block lengthofofn=2n=2mm --11 bitsbits..

The generator polynomial of this code can beThe generator polynomial of this code can bespecified in terms of its roots from the extensionspecified in terms of its roots from the extensionfield GF(2field GF(2mm))

The generator polynomialThe generator polynomial g(xg(x)) of theof the tt errorerrorcorrecting BCH code of lengthcorrecting BCH code of length 22mm --11 is the lowestis the lowestdegree polynomial over GF(2) which hasdegree polynomial over GF(2) which has

as its roots [as its roots [ ]]2 3 2, , ,....... t ( ) 0 1 2ig for i t =

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g(xg(x) has and their conjugates as all its r) has and their conjugates as all its rootsoots

IfIf

ThenThen

As everyAs everyeven powereven power ofof in the sequence ofin the sequence of

has the same minimal polynomial as sohas the same minimal polynomial as someme

preceding odd power ofpreceding odd power of in the sequence, the expressionin the sequence, the expression

for the generator polynomial reduces tofor the generator polynomial reduces to

2 3 2, , ,....... t

( ) is the minimal polynomial of ii x

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

2 3 2, , ,....... t

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

Since the degree of each minimal polynomial is

m or less, the degree ofg(x) is atmost mt.ie. The no. of parity bits,n k, will be atmost

mt.

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Single error correcting BCH code of lengthSingle error correcting BCH code of length

22mm--1 is generated by1 is generated by

g(xg(x) =) = 11(x)(x)

Since the primitive elementSince the primitive element specifies thespecifies the

primitive polynomialprimitive polynomial 11(x), single error(x), single error

correcting BCH code is a Hamming Code.correcting BCH code is a Hamming Code.

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Coding ProcedureCoding Procedure

The coding approach is different in BCH codeThe coding approach is different in BCH code

compared with othercompared with other LBCsLBCs

We begin by specifying the no. of random errorsWe begin by specifying the no. of random errors

we want the code to correct for a given blockwe want the code to correct for a given blocklength n.length n.

Then we construct the generator polynomial forThen we construct the generator polynomial forthat code.that code.

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TheBCHBound

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Definition: A binaryDefinition: A binarynn--tupletuple is a codewordis a codeword iffiffthethe

polynomialpolynomial

V(xV(x)= v)= v00+v+v11x+x+ .+v.+vnn--11xxnn--11 hashas ,, 22,, 2t2t asas

roots.roots.

This definition is useful in proving theThis definition is useful in proving the

minimum distanceminimum distance of the code.of the code.

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LetLetV(xV(x)= v)= v00+v+v11x+x+ .+v.+vnn--11xxnn--11 be a codebe a code

polynomial in apolynomial in a tt error correcting BCH code oferror correcting BCH code of

lengthlength n= 2n= 2mm--1.1.

SinceSince ii is a root ofis a root ofv(xv(x) for , then) for , then

This can be written as a matrix product asThis can be written as a matrix product as

1 2i t

2 ( 1)

0 1 2 1( ) ....... 0i i i n i

nv v v v v

= + + + + =

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2

( 1)

0 1 1

1

.

.

.

( , ,......., ) 0 .1 2

i

i

n i

n

v v v for i t

=

This means that the inner product of

is equal to zero. Now we can form the H matrix as

2 ( 1)

0 1 1( , ,......, ) (1, , ,...... )i i n i

nv v v and

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If for someIf for some i & j,i & j, jj is a conjugate ofis a conjugate ofii, then, then

v(v(jj)=0)=0 iffiffv(v(ii)=0)=0. This says that the inner. This says that the inner

product ofproduct ofv =(vv =(v00,v,v11,,.v.vnn--11)) and theand the ii--thth row ofrow of

H is zero, and the inner product ofH is zero, and the inner product ofvvand theand the jj--thth

row of H is also zero. So therow of H is also zero. So the jj--thth rowrowof H can beof H can beomitted.omitted.

Then theThen the HH matrix can be reduced to thematrix can be reduced to thefollowing form,following form,

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( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 3 1

2 3 13 3 3 3

2 3 15 5 5 5

2 3 12 1 2 1 2 1 2 1

1 . .

1 . .

1 . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

1 . .

n

n

n

nt t t t

H

=

The entries of H are elements from GF(2m). Each element in

GF(2m) can be represented by a m-tuple over GF(2). Then we get

the binary parity check matrix for the code.

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(15,7) BCH code(15,7) BCH code--Parity check matrixParity check matrix

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To prove the bound we have to show that noTo prove the bound we have to show that no 2t2tor fewer columns ofor fewer columns ofHH sums tosums to zero.zero. SupposeSuppose

that there exists a nonthat there exists a non--zero code vector of weightzero code vector of weight

LetLet

2t

1, 2, 3,( ..... ) be the nonzero components of vj j j jv v v v

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The second matrix on the right is a square matrix. To satisfy

the equality of the previous expression, the determinant of the

matrix must be zero.

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After taking out the common factor from eachAfter taking out the common factor from each

row of that determinant, we getrow of that determinant, we get

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The determinant in the above equality is aThe determinant in the above equality is a

VandermondeVandermondeDeterminantDeterminantwhich is nonwhich is non--zero.zero.

Hence the product on the RHS cannot be zero.Hence the product on the RHS cannot be zero.

This is a contradiction and hence our assumptionThis is a contradiction and hence our assumption

that there exists a codeword v of weight isthat there exists a codeword v of weight isinvalid.invalid.

This implies that the minimum weight of the tThis implies that the minimum weight of the terror correcting BCH code iserror correcting BCH code is atleastatleast 2t+1.2t+1.

2t

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2t+12t+1 is usually the designed distance of theis usually the designed distance of the tt--

error correcting BCH code. The true minimumerror correcting BCH code. The true minimum

distance of a BCH code may be greater than ordistance of a BCH code may be greater than or

equal to its designed distance.equal to its designed distance.