1. Reed-Solomon Codes
Rong-Jaye Chen

2. Reed-Solomon Codes [1] Codes over GF(2r) [2] Reed-Solomon codes
[3] Decoding Reed-Solomon codes
[4] Transform approach to Reed-Solomon codes (Omitted)
[5] Berlekamp-Massey algorithm
[6] Erasures (Omitted)

3. Reed-Solomon Codes [1] Codes over GF(2r) 1. GF(2r)[x]
Polynomials with coefficients from GF(2r)
K[x], K= GF(2)={0,1}, polynomials with binary coefficients
2. Recall BCH code of length n=2r-1
c(x)C iff 1, 2, 3, 4 are all roots of c(x), c(x)K[x]
1, 2, 4 share same minimal polynomial m1(x) and 3 has minimal polynomial m3(x) so the generator polynomial is gK(x)=m1(x)m3(x)

4. Reed-Solomon Codes 3. Generalization of BCH codes 4. Subfield subcode
Choosing c(x)GF(2r)[x]
c(x)C iff g(x)=(x+)(x+2)(x+3)(x+4) divides c(x)
4. Subfield subcode
CK is a BCH code
C over GF(2r) is an example of Reed-Solomon code
CK is said to be a subfield subcode of C:
CKC
All words in CK have all their digits in the subfield K
of GF(2r)  CK=C  Kn
Both CK and C are cyclic

5. Reed-Solomon Codes
5. If g(x) generates a linear cyclic code of length 2r-1 over GF(2r), the generator of the binary subfield subcode is the polynomial gK(x)
Let 1, 2,…, t be distinct non-zero elements of GF(2r). Then g(x)=(1+x) (2+x)…(t+x) generates a linear cyclic code of length 2r-1 over GF(2r)
(Theorem 6.1.1)
Example:
Let F= GF(24) constructed using 1+x+x4. g(x)=(+x) (2+x)= 3+5x+x2 generates a linear cyclic code over F of length 15. The codeword corresponding to g(x) is of course 3

6. Reed-Solomon Codes 6. Cyclic codes over GF(2r)
Let C be a linear cyclic code of length n over GF(2r). Every codeword c(x) can be written uniquely as m(x)g(x) for some m(x) in GF(2r)[x] of degree less than n-deg(g(x)). Also, g(x) divides f(x) iff f(x) is a codeword, and g(x) divides 1+xn
(Theorem 6.1.3)

7. Reed-Solomon Codes
Let g(x) have degree n-k. If g(x) generates a linear cyclic code C over GF(2r) of length n=2r-1 then
is a generating matrix for C.
|C|=2rk . There are 2rk polynomials m(x), since each of the k coefficients in m(x) can be any one of the 2r field elements

8. Reed-Solomon Codes Example:
Construct GF(23) using 1+x+x3 with  as the primitive element.
Let g(x)=(+x)(2+x)=3+4x+x2. Then g(x) generates a linear cyclic code C over GF(23) of length n=7 then
C has 85 codewords. The codeword corresponding to m(x)=1+x+3x41003=m, for example, is
m(x)g(x)  mG= 304613

9. Reed-Solomon Codes [2] Reed-Solomon codes 1. Lemma 6.2.1
Let 1, 2,…,t be non-zero elements of GF(2r). Then

10. Reed-Solomon Codes Example 6.2.2 we find that:
Using lemma and GF(24) constructed using 1+x+x4,
we find that:
=(7+2)(10+2)(10+7)
= 12 4 6
= 7

11. Reed-Solomon Codes 2. Theorem 6.2.4
Let g(x)=(m+1+x)(m+2+x)…(m+-1+x) be the
generator of a linear cyclic code C over GF(2r)
of length n=2r-1, where  is a primitive element
in GF(2r) and m is some integer. Then d(C)  
<Proof> any -1 rows in the parity check matrix H
(see p.131) is linearly independent
so d(C)  

12. Reed-Solomon Codes 3. Definition of RS(2r, ) 4. Theorem 6.2.5
A binary Reed-Solomon code RS(2r, ) is a cyclic code
over GF(2r) with generator g(x)=(m+1+x)(m+2+x)…
(m+-1+x) for some integer m and some primitive
element  of GF(2r)
4. Theorem 6.2.5
If C is an RS(2r, ) then
(a) n = 2r-1
(b) k = 2r-
(c) d = 
(d) |C| = 2rk

13. Reed-Solomon Codes 5. Binary representation of RS(2r, )
Any RS(2r, ) code C, can be represented as a
binary code simply by replacing each digit in
each codeword by the binary word
Example:
Let C be the RS(4,2) with g(x)= +x and where GF(22) is constructed using 1+x+x2. From Theorem 6.2.5, C has n=3, k=2, d=2 and |C|=16, a generating matrix for C is

14. Reed-Solomon Codes The 16 messages u with their binary representation
along with the corresponding codewords c = uG of C
and their binary representations

15. Reed-Solomon Codes 6. Shortened RS(2r, ) code C(s) Example namely:
Let C be the RS(4,2) code of previous example. The shortened RS(4,2) code C(1) is formed by taking all codewords that have 0’s in the last s=1 position,
namely:
000,10, 20 and 120
then deleting the last s positions. So
C(1)={ 00, 1, 2, 12 }

16. Reed-Solomon Codes Theorem 6.2.12 The distance d(s) of C(s):
d(C(s)) d(C) = 
d(s)  n(s) – k(s) + 1
= 2r-1-s-(2r--s)+1
= 
 d(s) = 
Theorem
Let C be an RS(2r, ) code and let C(s) be the shortened RS(2r, ) code with parameters n(s),k(s),and d(s), then
n(s) = 2r-1-s
k(s) = 2r--s
d(s) = 

17. Reed-Solomon Codes Example
In example we constructed an RS(23,3) code C with generator polynomial g(x)= 3+4x+x2. The shortened RS(23,3) code C(2) has generating matrix
and has parameters n(2)=5, k(2)=3 and d(2)=3. G(2) is formed by deleting the last s=2 rows columns of the original matrix G

18. Reed-Solomon Codes [3] Decoding Reed-Solomon codes
1. Error locations and error magnitudes
The error locations of a received word are the coordinates i in which the error pattern is non-zero, referred by an error location number I
The error magnitude of an error location i is the element of GF(2r) that occurs in coordinate i of the error pattern
Example 6.3.1: Using RS(8, 3) constructed in Example 6.1.5, if c= 3400000 is transmitted and w=3450000 is received then the most likely error pattern is c+w=e=00 So the error location number is 2 and the corresponding error magnitude is 4

19. Reed-Solomon Codes 2. Decoding algorithm Notation:
g(x)=(m+1+x)(m+2+x)…(m+-1+x)
t=(-1)/2
a1, …, ae = the error location numbers
b1, …, be = the error magnitudes

20. Reed-Solomon Codes Let w(x)=w0+w1x+ … + wn-1xn-1 Symdromes: s = wH
c(x)=c0+c1x+ … + cn-1xn-1
e(x)=e0+e1x+ … + en-1xn-1

21. Reed-Solomon Codes Syndrome: sm+1,…,sm+-1
sj=w(j) for m+1  j  m+-1
w(j)=c(j)+e(j)=e(j)= (6.1)
Error locator polynomial: A(x), let A={a1,…,ae}
A(x)=(a1-x) (a2-x) … (ae-x) (6.2)
Define j to be the coefficient of xj in A(x)
A(x)= 0+ 1x+ … + e-1xe-1+xe (6.3)
(6.4)
which may be written as
(6.5)

22. Reed-Solomon Codes
Substitute j=m+1,…,m+e in turn to obtain e linear equations
(6.6)
The exe matrix above be called M, which does have full rank

23. Reed-Solomon Codes
Equations (6.1) can most easily be represented as follows:
(6.7)
Define the M’ be the extended matrix:

24. Reed-Solomon Codes Algorithm 6.3.2
1. Calculate sj=w(j) for m+1  j  m+2t
2. Setting e=t, find the rank of the extended matrix M’
3. Let e be the rank of M’ and solve the linear system
(6.6) for 0,…,e-1
4. Find the roots of A(x)=0+ 1x+…+ e-1xe-1+xe
These roots are the error location numbers a1,…,ae
5. Solve the linear system (6.7) for b1,…,be
There are the error magnitudes corresponding to a1,…,ae

25. Reed-Solomon Codes Example 6.3.3
Let g(x)=(1+x)(+x)(2+x)(3+x)=6+5x+ 5x2+2x3+x4 be the generator of an RS(23,5) code (m=-1, t=2), where GF(23) is constructed using 1+x+x3. Suppose that the received word is w= 6  5 210 2
Using algorithm 6.3.2:
1. Since m=-1 and =5, we calculate the 4 syndromes
s0=w(0)=6++5+2+1+0+2=1
s1=w(1)=6+2+7+5+4+0+8=3
s2=w(2)=6+3+9+8+8+0+14=3
s3=w(3)=6+4+11+11+12+0+20=1

26. Reed-Solomon Codes Row reducing M’ yields the matrix which has rank 2
2. Setting e=t=2, the extended matrix M’ is
Row reducing M’ yields the matrix
which has rank 2

27. Reed-Solomon Codes
3. Since M’ has full rank. e=2 and so we now solve the linear system
However, as observed above, we have already row
reduced M in Step 2, so we have to solve
Then 41+ 2=0 so 1= 5, and 0+ 35 + 3 = 0
so 0= 1

28. Reed-Solomon Codes A(x) = 0+1x+x2=1+5x+x2. On substituting
4. We now know the error locator polynomial
A(x) = 0+1x+x2=1+5x+x2. On substituting
field elements into A(x), we find that A()=0
and A(6)=0. Therefore
A(x)=1+5x+x2 =(+x)(6+x)
So the error location numbers are a1=  and
a2= 6

29. Reed-Solomon Codes 5. Now we solve the following linear system: or
Then 5b2=1 so b2=2, and b1+b2=1 so b1=6.
Therefore the most likely error pattern is and
the most likely codeword is
c = w+e = 65 52100

30. Reed-Solomon Codes Example 6.3.4 Let
be the generator of an RS(24,7) code (so m=-1 and t=3), where GF(24) is constructed using
(see Table 5.1)
Suppose that the received word is

31. Reed-Solomon Codes 1.

32. Reed-Solomon Codes 2.
So M has rank 2 and therefore the most likely error
pattern has weight e=2

33. Reed-Solomon Codes 3. With e=2, the linear system (6,7) becomes
but in Step 2 we row reduced this matrix:
Then 12 1+7=0 so 1=10, and 7 0+1+9=0
so 0=6.

34. Reed-Solomon Codes and a2=4 5. so
4. A(x)=6+ 10x+x2=(2+x)(4+x). Therefore a1= 2
and a2=4 5. so
Therefore b2= 12 and b1= 2. So the most likely error
pattern is e=0020120…0 and the most likely codeword is
c=w+e=142129100…0

35. Reed-Solomon Codes
[4] Transform approach to Reed-Solomon codes (Omitted)
[5] Berlekamp-Massey algorithm
1. Notation
Reverse error locator polynomial
R(x)=1+t-1x+t-2x2+…+0xt
Syndrome polynomial
s(x)=1+sm+1x+sm+2x2+…+sm+2tx2t
we can write
R(x)s(x)=q(x)x2t+1+r(x) with deg(r(x))  t

36. Reed-Solomon Codes 2. Algorithm 6.5.1
Produce a sequence of polynomials Pi(x) and integer Di 
If Pi(x)s(x)=qi(x)xi+1+ri(x) with deg(ri(x))  i ,
then deg(Pi(x))  i -  Di /2 and deg(ri(x))  i - (1+Di)/2
Pi(x) is a combination of Pi-1(x) and some specific previous polynomial Pz(i-1)(x)
2. Algorithm 6.5.1
Calculate sj=w(j) for m+1  j  m+2t
Define:
q-1(x)=1+sm+1x+sm+2x2+…+sm+2tx2t
q0(x)=sm+1+sm+2x+…+sm+2tx2t-1
p-1(x)=x2t+1
p0(x)=x2t
Let D-1=-1 and D0=0 and let z0=-1

37. Reed-Solomon Codes
For 1  i  2t, recursively define qi(x), pi(x), Di, and zi
as follows
(a) If qi-1,0=0, then let
qi(x)= qi-1(x)/x
pi(x)= pi-1(x)/x
Di= 2+Di-1
zi= zi-1
(b) If qi,0  0, then let

38. Reed-Solomon Codes and let
which can be truncated to have degree at most 2t-1-i;
and let
If e  t errors have occurred during transmission then
p2t(x)= R(x) has degree e; the error locator polynomial
is:
which has e distinct roots

39. Reed-Solomon Codes Example 6.5.2:
Consider example with the syndrome s0=7,
s1=0, s2=9, s3=12, s4=9, s5=7. Working step by
step through Algorithm 6.5.1, we obtain the following.
We begin by setting
q-1(x) = 1+7x+x2+9x3+12x4+9x5+7x6
q0(x) = 7+x+9x2+12x3+9x4+7x5
p-1(x) = x7
p0(x) = x6
D-1 = -1, D0= 0, z0= -1

40. Reed-Solomon Codes (q0(x)+7q-1(x))/x =3+x+13x2+14x3+14x4+14x5
Let i=1. Since q0,0=70, we use step 3(b):
(q0(x)+7q-1(x))/x
=3+x+13x2+14x3+14x4+14x5
which is truncated to degree 2t-i-1=4 to give
q1(x) =3+x+13x2+14x3+14x4
p1(x) =1+7x
D1 = 2+min{D-1, D0}=0
and since D0D-1, zi=i-1=0
Before proceeding, we shall adopt a more concise format by representing the polynomials by their corresponding words. Then the information we have found so far is the following table:

41. Reed-Solomon Codes i qi - pi Di zi -1 0 7 0 9 12 9 7 - 0 -1
-1 0 7 0 9 12 9  0 -1
0 7 0 9 12 9  
1 3 0 13 14  0 
Proceeding from i=2 to i=2t=6 we obtain the following table
2 12 7 6  0 
3 0 10  0 12 
0 10 
0 10 
0 10 
So finally we obtain (x) by reading p2t(x)= p6(x)= backwards:
(x) = 6 + 10x+ x2

42. Reed-Solomon Codes Example 6.5.3:
Let C be the RS(24,9) code with generator g(x)=(1+x)(+x)… (7+x) and GF(24) constructed using 1+x+x4. Suppose that w is the received word and the syndromes of w are:
s0=12, s1=9, s2=6, s3=3, s4=5, s5=12,s6=6, s7=6
Using algorithm and the notation describe in Example 6.5.2, we obtain the error locator polynomial as follows:
i qi pi Di zi
-1 0 12 9 6 3 5 12 6  0 -1
0 12 9 6 3 5 12 6  
10 7 5  0 
10 7 5  0 
10 7 5  0 
4 10 7 5  0 
8  0 12 0 0 
6 8  0 12 0 0 
0 12 13 10 
0 12 13 10 
Therefore the error locator polynomial is:
(x) = 13 + 10x+ 13x2+12x3+x4

43. Reed-Solomon Codes [6] Erasures (Omitted)
3. Error evaluator polynomial
Since r2t(x)=p2t(x)q-1(x) mod x2t+1
Either r(x)=r2t(x)
or (x)=r2t(x)-p2t(x)
called the error evaluator polynomial, used to calculate the error magnitudes bj, given the error locations aj
Example 6.5.2
[6] Erasures (Omitted)