1. ADVANTAGE of GENERATOR MATRIX:
we need to store only the k rows of G instead of 2k vectors of
the code.
for the example we have looked at generator array of (36)
replaces the original code vector of dimensions (8  6).
This is a definite reduction in complexity .
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2. Systematic Linear Block Codes
Systematic (n,k) linear block codes has such a mapping that
part of the sequence generated coincides with the k message
digits.
Remaining (n-k) digits are parity digits
A systematic linear block code has a generator matrix of the form :
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3. P is the parity array portion of the generator matrix pij = (0 or 1)
Ik is the (kk) identity matrix.
With the systematic generator encoding complexity is further
reduced since we do not need to store the identity matrix.
Since U = m G
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4. given the message k-tuple m= m1,……..,mk
where
And the parity bits are
given the message k-tuple
m= m1,……..,mk
And the general code vector n-tuple
U = u1,u2,……,uk
Systematic code vector is:
U = p1,p2……..,pk , m1,m2,……,mk
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5. For a (6,3) code the code vectors are described as
Example:
For a (6,3) code the code vectors are described as
P I3
U = m1+m3, m1+m2, m2+m3, m1, m2, m3
u1 , u2 , u3, u4, u5, u6
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6. Parity Check Matrix (H)
We define a parity-check matrix since it will enable us to
decode the received vectors.
For a (kn) generator matrix G
There exists an (k-n) n matrix H
Such that rows of G are orthogonal to the rows of H
i.e G HT = 0
To satisfy the orthogonality requirement H matrix is written as:
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7. The product UHT of each code vector is a zero vector.
Hence
The product UHT of each code vector is a zero vector.
Once the parity check matrix H is formed we can use it to
test whether a received vector is a valid member of the
codeword set. U is a valid code vector if and only if
UHT=0.
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8. Syndrome Testing
Let r = r1, r2, ……., rn be a received code vector (one of 2n n-tuples)
Resulting from the transmission of U = u1,u2,…….,un (one of the 2k n-tuples).
 r = U + e
where e = e1, e2, ……, en is the error vector or error pattern introduced by the channel
In space of 2n n-tuples there are a total of (2n –1) potential nonzero error patterns.
The SYNDROME of r is defined as:
S = r HT
The syndrome is the result of a parity check performed on r to determine whether r is a valid member of the codeword set.
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9. If r contains detectable errors the syndrome has some non-zero value
 syndrome of r is seen as
S = (U+e) HT = UHT + eHT
since UHT = 0 for all code words then :
S = eHT
An important property of linear block codes, fundamental to the
decoding process, is that the mapping between correctable error
patterns and syndromes is one-to-one.
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10. Parity check matrix must satisfy:
No column of H can be all zeros, or else an error in the
corresponding code vector position would not affect the
syndrome and would be undetectable
2. All columns of H must be unique. If two columns are identical errors corresponding to these code word locations will be indistinguishable.
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11. Example: P is the parity matrix and I is the identity matrix.
Suppose that code vector U = [ ] is transmitted and the vector r = [ ] is received.
Note one bit is in error..
Find the syndrome vector,S, and verify that it is equal to eHT.
(6,3) code has generator matrix G we have seen before:
P I
P is the parity matrix and I is the identity matrix.
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12. (syndrome of corrupted code vector)
S = r HT =
= [ 1, 1+1, ] = [ ]
(syndrome of corrupted code vector)
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13. ( =syndrome of error pattern )
Now we can verify that syndrome of the corrupted code vector is the same as the syndrome of the error pattern:
S = eHT = [ ]HT = [ ]
( =syndrome of error pattern )
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14. Error Correction
Since there is a one-to-one correspondence between correctable error patterns and syndromes we can correct such error patterns.
Assume the 2n n-tuples that represent possible received vectors are arranged in an array called the standard array.
1. The first row contains all the code vectors starting with all-zeros
vector
2. First column contains all the correctable error patterns
The standard array for a (n,k) code is:
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15. The array contains 2n n-tuples in the space Vn
Each row called a coset consists of an error pattern in the first column, also known as the coset leader, followed by the code vectors perturbed by that error pattern.
The array contains 2n n-tuples in the space Vn
each coset consists of 2k n-tuples
 there are cosets
If the error pattern caused by the channel is a coset leader, the received vector will be decoded correctly into the transmitted code vector Ui. If the error pattern is not a coset leader the decoding will produce an error.
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16. Syndrome of a Coset If ej is the coset leader of the jth coset then ;
Ui + ej is an n-tuple in this coset
Syndrome of this coset is:
S = (Ui + ej)HT = Ui HT + ejHT
= ejHT
All members of a coset have the same syndrome and in fact the syndrome is used to estimate the error pattern.
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17. Error CorrectionDecoding
The procedure for error correction decoding is as follows:
Calculate the syndrome of r using S = rHT
Locate the coset leader (error pattern) , ej, whose syndrome equals rHT
This error pattern is the corruption caused by the channel
The corrected received vector is identified as U = r + ej .
We retrieve the valid code vector by subtracting out the identified error
Note: In modulo-2 arithmetic subtraction is identical to that of addition
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18. Example: Locating the error pattern:
For the (6,3) linear block code we have seen before the standard array can be arranged as:
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19. The valid code vectors are the eight vectors in the first row and the correctable error patterns are the eight coset leaders in the first column.
Decoding will be correct if and only if the error pattern caused by the channel is one of the coset leaders
We now compute the syndrome corresponding to each of the correctable error sequences by computing ejHT for each coset leader
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20. Syndrome lookup table.. error pattern Syndrome 0 0 0 0 0 0 0 0 0
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21. Û = r + ê = (U + e) + ê = U + (e ê)
Error Correction
We receive the vector r and calculate its syndrome S
We then use the syndrome-look-up table to find the corresponding error pattern.
This error pattern is an estimate of the error, we denote it as ê
The decoder then adds ê to r to obtain an estimate of the transmitted
code vector û
Û = r + ê = (U + e) + ê = U + (e ê)
If the estimated error pattern is the same as the actual error pattern that is if ê = e then û = U
If ê  e the decoder will estimate a code vector that was not transmitted and hence we have an undetectable decoding error.
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22. Example
Assume code vector U = [ ] is transmitted and the vector r=[ ] is received.
The syndrome of r is computed as:
S = [ ]HT = [ ]
From the look-up table 100 has corresponding error pattern:
ê = [ ]
The corrected vectors is the Û = r + ê =
= (corrected)
In this example actual error pattern is the estimated error pattern,
Hence û = U
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