Chapter 5 BCH Codes

Chapter 5
BCH Codes
1. Introduction
BCH (Bose – Chaudhuri - Hocquenghem) Codes form a large
class of multiple random error-correcting codes.
They were first discovered by A. Hocquenghem in 1959 and
independently by R. C. Bose and D. K. Ray-Chaudhuri in 1960.
BCH codes are cyclic codes. Only the codes, not the decoding
algorithms, were discovered by these early writers.
The original applications of BCH codes were restricted to binary
m
codes of length 2 − 1
for some integer m. These were extended
later by Gorenstein and Zieler (1961) to the nonbinary codes with
symbols from Galois field GF(q).

The first decoding algorithm for binary BCH codes was devised
by Peterson in 1960. Since then, Peterson’s algorithm has been
refined by Berlekamp, Massey, Chien, Forney, and many others.
2. Primitive BCH Codes
For any integer m ≥ 3
and
t < 2 m−1
there exists a primitive
BCH code with the following parameters:
m
n = 2 − 1
n − k ≤ mt
dmin ≥ 2t
+ 1
(5- 1)
This code can correct t or fewer random errors over a span of
m
2 − 1
bit positions.
Thecodeisat-error-correcting BCH code.
For example, for m=6, t=3
6
n = 2 − 1 = 63
n − k = 6 × 3 = 18
d = 2×
3 + 1 = 7
min
This is a triple-error-correcting (63, 45) BCH code.

3. Generator Polynomial of Binary BCH Codes
m
Let α be a primitive element in GF( 2 ).
For
1 ≤ i ≤ t ,let φ )
( 2i−1 x
be the minimum polynomial of the
field element
2i−1
α .
Thedegreeof φ2i−1(
x)
is m or a factor of m.
The generator polynomial g(x) ofat-error-correcting primitive
m
BCH codes of length 2 − 1
is given by
{ ( x),
φ ( x),
, ( )}
g( x)
= LCM φ1 3
L φ2t−1
x
(5- 2)
Note that the degree of g(x)ismt or less.
Hence the number of parity-check bits; n-k, of the code is at
most mt.
Example 5.1, 5.2, 5.3, 5.4. (pp. 191-196)
(m =4,m =5)

Note that the generator polynomial of the binary BCH code is
originally found to be the least common multiple of the minimum
polynomials
φ , φ
1
φ
2,
L,
2t
i.e. g x)
= LCM{ φ ( x),
φ ( x),
φ ( x),
L,
φ ( x),
φ ( )}
(
1 2 3
2t−1
2t
x
However, generally, every even power of α
m
in GF( 2 )hasthesame
minimal polynomial as some preceding odd power of α
2m
in GF( 2 ).
As a consequence, the generator polynomial of the t-error-correcting
binary BCH code can be reduced to
{ ( x),
φ ( x),
, ( )}
g( x)
= LCM φ1 3
L φ2t−1
x .

Example: m =4,t =3
Let α
4
be a primitive element in GF( 2 ) which is constructed
based on the primitive polynomial
p ( x)
= 1 + x + x
4
φ ( x + x + x
1
) = 1
4
corresponding to α
φ
2 3 4
3
( x ) = 1 + x + x + x + x
corresponding to
3
α
φ ( x + x + x
5
) = 1
2
corresponding to
5
α
{ φ ( x),
φ ( x),
φ ( x)
}
g(
x)
= LCM
1 3
= φ1(
x)
φ3
( x)
φ5
( x)
2 4
= 1 + x + x + x + x
5
5
+ x
The code is a (15, 5) cyclic code.
8
+ x
10

4. Properties of Binary BCH Codes
Consider a t-error-correcting BCH code of length n = 2 m − 1
with generator polynomial g(x).
g(x) has as
2 3 2t
α, α , α , L,
α roots, i.e.
i
g( α ) = 0 for 1 ≤ i ≤ 2t
(5-
3)
Since a code polynomial c(x) is a multiple of g(x), c(x) alsohas
2 2t
i
α, α , L,
α as roots, i.e. c( ) = 0 for 1 ≤ i ≤ 2t
α .
m
A polynomial c(x)ofdegreelessthan 2 − 1
is a code polynomial
ifandonlyifithas
2 2t
α, α , L,
α as roots.

5. Decoding of BCH Codes
Consider a BCH code with n = 2 m − 1
and generator polynomial
g(x).
Suppose a code polynomial
n−1
c(
x)
= c0
+ c1x
+ L+
c n −1x
is
transmitted.
Let
n−1
r(
x)
= r0
+ r1
x + L+
r n −1x
be the received polynomial.
Then r(x)=c(x)+e(x), where e(x) is the error polynomial.
To check whether r(x) is a code polynomial or not, we simply test
2
2t
whether r(
α)
= r(
α ) = L = r(
α ) = 0 .
If yes, then r(x) is a code polynomial, otherwise r(x) is not a code
polynomial and the presence of errors is detected.
Decoding procedure
(1) syndrome computation.
(2) determination of the error pattern.
(3) error correction.

6. Syndrome computation
m
The syndrome consists of 2t components in GF( 2 )
S
= S S L S )
(5-
(
1 2 2t
4)
i
and S = r(
α ) for 1 ≤ i ≤ 2t
.
i
Computation:
Let φ (x)
i
be the minimum polynomial of
i
α .
Dividing r(x)by φ (x)
,weobtain
r( x)
= a(
x)
φ
i
( x)
+ b(
x)
i
i
Then S = b(
α )
(5- 5)
i
i
S = b(
α ) can be obtained by linear feedback shift-register with
i
connection based on φ (x)
.
i

7. Syndrome and Error Pattern
Since r(x)=c(x)+e(x)
i
i
i
i
then s = r(
α ) = c(
α ) + e(
α ) = e(
α )
(5-
6)
i
for
1 ≤ i ≤ 2t
.
This gives a relationship between the syndrome and the error
pattern.
Suppose e(x) hasν errors ( ν ≤ t)
at the locations specified by
j j2
x , x , ,
1 L j
x ν
.
i.e.
e = L+
j1
j2
j
( x)
x + x + x ν
(5- 7)
where ≤ j < j < L < j ≤ n 1 .
0
1 2
ν
−
From equations (5-6) & (5-7), we have the following relation
between syndrome components and error location:
S
S
S
1
2
2t
j1
= e(
α)
= α
j2
+ α
jν
+ L + α
2 j1
2
= e(
α ) = ( α )
j2
2
+ ( α )
jν
2
+ L+
( α )
M
2t
j1
2t
= e(
α ) = ( α )
j2
2t
+ ( α )
jν
+ L + ( α )
2t
(5- 8)

It we can solve the 2t equations, we can determine
j
α
1 j2
jν
, α , L , α .
ju The unknown parameter α = Z
u for u = 1,2,
L,
ν
are called
the “error location number”.
When
1 are found, the powers j u ,
ju
α , ≤ u < ν
u = 1,2,
L,
ν
give us the error locations in e(x). these 2t equation of eq. (5-8)
Are known as power-sum symmetric function.

8. Error-Location Polynomial
(Error-Locator Polynomial)
suppose that ν ≤ t
errors actually occur.
Define error-locator polynomial L(z)as
L(
z)
= (1 + Z z)(1
+ Z z)
L(1
+ Z
=
ν
∏
i=
1
(1 + Z z)
= σ + σ z + σ z
0
1
1
i
2
2
2
+ L+
σ z
ν
ν
ν
z)
(5- 9)
where σ = 0
1 .
L(z)has
Z L as roots.
−1
−1
−1
1
, Z2
, , Z ν
Note that
Z
u
j
= α u
.
If we can determine L(z) from the syndrome S = S , S , L,
S ) ,
(
1 2 2t
then the roots of L(z) give us the error-location numbers.

9. Relationship between S and L(z)
From eq. (5-9), we find the following relationship between the
coefficients of L(z) and the error-locator numbers:
σ
σ
1
σ
σ
2
ν
=
= Z
= Z
M
= Z
0
1
1
1
1
+ Z2
+ L+
Zν
Z + Z Z + L+
Z
Z
2
2
2
LZ
ν
3
Z
ν − 1 ν
(5- 10)
eq. (5-10) is called “elementary symmetric functions”.
From eq. (5-8) and eq. (5-10), we have the following relationship
between the syndrome and the coefficients of L(z):
S
ν
+ σ S
1
ν −1
+ σ S
2
S
3
ν −2
S
+ σ S
1
2
2
+ L+
σ
S1
+ σ
1
+ σ
1S1
+ 2σ
2
+ σ S + 3σ
2
ν −1
S
1
1
+
νσ
3
ν
= 0
= 0
= 0
M
= 0
(5- 11)
Here for binary case
iσ = σ
i
i
when i is odd,
and iσ = 0
i
otherwise.
The equations of (5-11) are called the Newton’s identities.

If we can determine σ , σ , 1 2
L,
σ
ν from the Newton’s identities,
then we can determine the error-location numbers
Z , Z , 2
, Z
1
L
ν ,
by finding the roots of L(z).
Note that the Newton’s identities in (5-11) can be expressed also in
the following single-equation form:
S
i
+
1 i−1
2 i−2
i−1
1 i
L
σ S + σ S + L + σ S + iσ
= 0 for i = 1,2, , ν
(5- 12)

10.Peterson’s Direct-Solution (W. W. Peterson, 1960)
Considerthecasefor i > ν
First multiply L(z) in eq. (5-9) By
i
z − , we have
z
−i
L
−i
1−i
2−i
ν −1−i
ν −i
( z)
= z + σ
1z
+ σ
2z
+ L + σ ν −1z
+ σ
ν
z (5- 13)
Next substituting the roots of L(z) (i.e.
Z L ) into eq.
−1
−1
−1
1
, Z2
, , Z ν
(5-13) produces the following set of equations:
Z
Z
Z
i
1
i
2
i
ν
+ σ
1Z
+ σ Z
1
+ σ Z
1
i−1
1
i−1
2
i−1
ν
+ σ
2
Z
+ σ Z
2
+ σ Z
2
i−2
1
i−2
2
i−2
ν
+ L + σ
+ L + σ
+ L + σ
ν −1
ν −1
ν −1
Z
Z
Z
i−ν
+ 1
1
i−ν
+ 1
2
i−ν
+ 1
ν
+ σ
ν
Z
+ σ Z
ν
+ σ Z
ν
i−ν
1
i−ν
2
i−ν
ν
= 0
= 0
M
= 0
(5- 14)
Adding these ν
equation term by term yield
( Z
i
1
+ Z
i
2
+ L+
Z
i
ν
i−1
i−1
) + σ
1(
Z1
+ Z
2
+ L+
Z
i−ν
i−ν
+ σ ( Z , Z , L,
Z
ν
1
2
i−1
ν
i−ν
ν
) + L
) = 0
(5- 15)
Now express eq. (5-15) in terms of syndrome components, then
S
i
+
−
σ
1Si−1
+ L + σ
ν −1Si−ν
+ 1
+ σ
ν
Si
ν
= 0 for i > ν (5- 16)
In particular, for i =ν + 1 ,weobtain
S
ν + 1
+
1 ν
ν −1
2 ν
S1
=
σ S + L + σ S + σ 0
(5- 17)
Thus, the Newton’s identities can be extended to the unknown
syndrome
S for i > ν .
i

From eq. (5-16) & eq. (5-17), we can see that the σ
j
for
0 ≤ j ≤ν are closely related to the syndrome components S
i ,
1 ≤ i ≤ν
+ 1 .
Thus, σ
j
can be determined by solving the set of syndrome
equations eq. (5-16) & eq. (5-17). Then the error-locations for can
be found by solving
ju α = Z
u for ≤ u ≤ν
1 the root of L(z). This
L(z) produces an error-pattern e(x) with the minimum number of
errors. Hence if
ν ≤
t
errors occur, L(z) will give the actual error
pattern e(x).
Finally, the error-correcting procedure for the binary BCH codes
can be outlined as follows:
(1) Compute the syndrome components S
j ,
1 ≤ j ≤ 2t
,fromthe
received polynomial r(x). [eq. (5-5)]
(2) Set each σ = 0 for ν + 1 ≤ j ≤ t
j
and solve the first ν equations of eq. (5-11) for the σ
j ,
1 ≤ j ≤ν in terms of S
j .
(3) Determine the error-locator polynomial L(z) fromthese σ
j

in terms of syndrome component
S for 0 ≤ j ≤ 2t
.
j
(4) Find the error-location numbers Z , Z , 1 2
L,
Zν
by solving
for the roots of L(z). Use these roots to correct the errors in
r(x).

11.Direct Solutions of Some Simple Cases.
(1) Single-error correction:
σ
1
= S1
L(
z)
= 1 + S1z
S1
≠ 0
3
S3
= S1
(2) Double-error correction:
σ
σ
1
2
= S
1
= S
−1
1
( S
3
+ S
3
1
)
3
S1
≠ 0, S3
≠ S1
3
⎡ S1
+ S
L(
z)
= 1 + S1z
+ ⎢
⎣ S
1
3
⎤
⎥z
⎦
2
(3) Triple-error correction:
σ
σ
σ
1
2
3
= S
S
=
= ( S
1
2
1
S3
3
S1
3
1
+ S
3
5
+ S3
+ S ) + S σ
1
2
S
S
1
3
≠ 0
≠ S
3
1
4
Example: m =4,t = 3 BCH code over GF( 2 ).
The primitive polynomial for m =4is
φ( x ) = 1+
x + x
4
The minimum polynomials of α ,
3
α
and
5
α
are
φ1(
x)
= 1 + x + x
φ3(
x)
= 1 + x + x
φ ( x)
= 1 + x + x
5
4
2
2
+ x
3
+ x
4
n = 2 4 − 1 =
15
g(
x)
= LCM{
φ1(
x),
φ
3(
x),
φ5
( x)}
= φ1φ
3φ5
2 4 5 8
= 1 + x + x + x + x + x + x
10

The code is a (15, 5) cyclic code.
The generator polynomial g(x) has
α
2 3 4 5 6
, α , α , α , α , α
as
roots. The roots α ,
2
α
and
4
α
have the same polynomial
φ ( x + x + x
1
) = φ2(
x)
= φ4
( x)
= 1
4
The root
3
α
and
6
α
have the same minimum polynomial
φ
2 3 4
3( x ) = φ6
( x)
= 1 + x + x + x + x
The minimum polynomial of
5
α
is
φ ( x + x + x
5
) = 1
2
Suppose all-zero code word c = ( 0 0 0L0)
is transmitted
and
r ( x)
+ x
2 5 12
= x + x is received.
Dividing r(x)by φ ( ) , φ ( ) and φ ( ) , respectively,
1 x
we obtain the remainders:
b1
( x)
= 1
b3
( x)
= 1 + x + x
2
b ( x)
= x
5
3
3 x
5 x
The syndrome components are
s
s
s
s
s
s
1
2
4
3
6
5
= b1
( α)
= 1
2
= b1
( α ) = 1
4
= b1
( α ) = 1
3
6 9 10
= b3
( α ) = 1 + α + α = α
6
12 18 5
= b3
( α ) = 1 + α + α = α
5 10
= b ( α ) = α
3
10 10 5
Hence S = (1 , 1 , α , 1 , α , α ) .

Example: (p. 209) consider t =2,m = 5, (32, 21) BCH code
The primitive polynomial for m =5is
φ
2 5
( x ) = 1 + x + x
α
2 4 8
, α , α , α
and
16
α
have the same minimum polynomial
α
3 6 12 24
, α , α , α
and
17
α
have the same minimum polynomial
φ
2 3 4 5
3( x ) = 1 + x + x + x + x
α
5 10 20 9
, α , α , α
and
8
α
have the same minimum polynomial
φ
2 4 5
5( x ) = 1 + x + x + x + x
α
7 14 28 25
, α , α , α
and
19
α
have the same minimum polynomial
φ
2 3 5
7
( x ) = 1 + x + x + x + x
α
11 22 13 26
, α , α , α
and
21
α
have the same minimum polynomial
φ
3 4 5
11( x ) = 1 + x + x + x + x
The generator polynomial of (32, 21) code is
g +
3 5 6 8 9 10
( x)
= φ1(
x)
φ3(
x)
= 1 + x + x + x + x + x x
The roots of the generator polynomial include
α
2 3
, α , α
and
4
α
Since
r +
2 7 8 11 12
( x)
= x + x + x + x x

We obtain
S
S
S
S
1
2
3
4
7
= r(
α)
= α
2
= r(
α ) = α
3 8
= r(
α ) = α
4
= r(
α ) = α
14
28
Note here that
S =
2 4
4
= ( S2
) ( S1
)
Then the error-locator polynomial is obtained by the equation
7
σ
1
= α
8 7 3
α + ( α ) 15
σ
2
= = α
7
α
7 15
∴L(
z)
= 1 + α z + α z
5 31−26
∴ Z1
= α = α
10 31−21
Z = α = α
2
2
5
10
= (1 + α z)(1
+ α z)
Thereby indicating errors at 26 th and 21 st coordinates of r
∴c +
2 5 7 8 10 11 12
( x)
= x + x + x + x + x + x x
(Note
Zi
r
= α i
)

12.Computation of Error-Location Numbers
– Chien Search. (R.T. Chien, 1964)
A Chien-search circuit is shown in Fig. 5.2.
L ( z ) = σ
where σ 0
= 1
ν
ν
+ σ z + σ z 2
+ L + σ z
0 1 2
ν
= ∏(1
+ Z z i
)
i=
1
m
The roots of L(z) inGF( 2 ) can be determined by substituting
m
the elements of GF( 2 )inL(z).
i
If L( α ) = 0 ,then
i
α
is the root of L(z) and
α
− i
= α
n−i
is an
error-location number.
To decode the first received digit r
n−1
, we check whether α is a
root of L(z). If L ( α)
= 0 , then rn−
1
is erroneous and must be
corrected.
If L ( α)
≠ 0 ,then rn−
1
is error-free.
i
i
To decode rn
− i ,wetestwhether L( α ) = 0 .if L( α ) = 0 , n i
r −
is
erroneous and must be corrected, otherwise
rn
− i
is error free.

= ( r ,r , L,
1)
0 1
rn-
c'
c'
c'
e i
e
i
⎧ 1, if Di
= 1
= ⎨
⎩0,
otherwise
D
i
t
ij
= ∑σ
jα
j=
1
i
= L(α
) − 1
∑
σ 1
σ
2
L
σ
t
2
α α
t
α
Figure 5.2 Chien search and error-correction for binary code.

13.Peterson-Gorenstein-Zierler Decoding Algorithm
From equation (5-16) [Chen / Reed eq. (5-38)]
σ
ν
S
j−ν
+ σ
ν −
S
j−ν
+
+ + σ S
j−
= −S
j
for j > ν
1 1
L
1
1
Assuming that
ν =
t ,then
σ S σ S + L + S = −S
for j t (5- 18)
t j−t
+
t−1 j−t+
1
σ
1 j−1
j
>
e.g. j = t + 1, we have
σ S
t
1
+ σ
t−1S2
+ L+
σ
1St
= −St+
1
The following matrix equation is obtained for the symmetric
function
σ
j
as follows:
S'
Λ =
⎡S
⎢
S
⎢
⎢ M
⎢
⎣S
1
2
t
S
S
S
M
2
3
t+
1
L
L
S
S
t−1
S
M
t
2t−2
S
S
S
t
t+
1
M
2t−1
⎤⎡
σ
t
⎥⎢
σ
⎥⎢
t−
⎥⎢
M
⎥⎢
⎦⎣
σ
1
1
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡−
⎢
−
⎢
⎢
⎢
⎣ −
S
S
M
S
t+
1
t+
2
2t
⎤
⎥
⎥
⎥
⎥
⎦
(5- 19)
It can be shown that S'
is nonsingular if the received word
contains exactly t errors. It also can be shown that
S'
is singular
if fewer than t errors occur. If
S'
is singular, then the rightmost
column and bottom rows can be removed and the determinant of
the resulting matrix computed.

This process is repeated until one reaches a non-singular matrix.
The coefficients of the error-locator polynomial are then founded
by the use of standard algebraic technique.
Once the ν error locations are known, hen we can use the
relation between syndrome components and error locations.
[eq. (5-8)]
The syndrome components can be computed by
S
j
j
= e(
α ) =
n−1
∑
k=
0
e
k
j
⋅ ( α )
k
=
ν
∑
l=
1
e
i
l
⋅ Z
j
l
where
e(
x)
= e
0
+ e
1
x + e
2
x
2
+ L+
e
n
−
∑ − 1
n 1
n− 1
x =
k=
0
e
k
x
k
Thus we have
S
S
S
1
2
2t
= e
i
= e
M
= e
i
i
1
1
1
Z
Z
Z
1
2
1
2t
1
+ e
i
+ e
2
i
2
+ e
i
Z
2
Z
2
2
2
Z
+ L+
e
2t
2
+ L + e
+ L+
e
i
ν
i
ν
Z
i
ν
2
ν
Z
ν
Z
2t
ν

The system equations can be reduced to the following matrix
form:
D ⋅ e
⎡ Z1
⎢ 2
Z
= ⎢
1
⎢ M
⎢
⎣Z1
ν
Z
Z
M
Z
2
2
2
ν
2
L
L
L
Z
Z
Z
ν
2
ν
M
ν
ν
⎤⎡ei
⎥⎢
e
⎥⎢
i
⎥⎢
M
⎥⎢
⎦⎣
ei
1
2
ν
⎤
⎥
⎥ =
⎥
⎥
⎦
⎡S
⎢
S
⎢
⎢ M
⎢
⎣S
1
2
ν
⎤
⎥
⎥
⎥
⎥
⎦
(5- 20)
Decoding is completed by solving for the { e il
}
This is a general case of nonbinary BCH codes.
Example (5.9)
Consider a nonbinary BCH code (7, 3) of length 7 (symbols) this
code is constructed over GF(8) with generator polynomial
2
3
4
g(
x)
= ( x − α)(
x − α )( x − α )( x − α )
4 3 3 2 3
= x + α x + x + α
Let the received polynomial be
r
2 6 2 4 3 5 2
( x)
= α x + α x + x + α x
Then the syndrome components are
S
3
= α , S = S = S = α
6
3
4
1 2
α ,
3
α ,
4

Eq. (5-19) gives
S' Λ =
6
⎡α
⎢ 3
⎣α
3
α ⎤⎡σ
2 ⎤
4 ⎥⎢
α σ
⎥
⎦⎣
1 ⎦
=
4
⎡α
⎤
⎢ 3 ⎥
⎣α
⎦
⇒ σ
1
2
= α ,
σ
2
= α
2 2
Thus L(
x)
= αx
+ α x + 1
The error locations are founded to be
Z
1
3
= α
= α
7−4
,
Z
2
5
= α
= α
7−2
⇒ e(
x)
= e
3
x
3
+ e
5
x
5
Eq. (5-20) gives
D
3
⎡α
= ⎢ 6
⎣α
5
α ⎤⎡e
3 ⎥⎢
α ⎦⎣e
6
⎤ ⎡α
⎤
⎥ = ⎢ ⎥
⎦ ⎣α
⎦
3
⋅ e
3
5
The error magnitudes are found to be
e
3
= α,
e
5
5
= α
⇒ e(
x)
= αx
3
5
+ α x
5
Finally
c
2 6 5 5 2 4 3 3 5 2
( x)
= e(
x)
+ r(
x)
= α x + α x + α x + α x + α x
Note that m = 3, the primitive polynomial is
p ( x)
= 1 + x + x
3

14.BCH Codes as Industry Standards
(a) (511, 493) BCH code in ITU-T. Rec. H.261 “video codec for
audiovisual service at kbit/s” a video coding a standard used for
video conferencing and video phone.
n = 511
k = 493
m = 9
n − k = 18
t = 2
(b) (40, 32) BCH code in ATM (Asynchronous Transfer Mode) pp.
223-227.
This is shortened cyclic code that can correct 1-bit error and
detect 2-bit errors.