1. 3F4 Error Control Coding
Dr. I. J. Wassell

2. Introduction Error Control Coding (ECC)
Extra bits are added to the data at the transmitter (redundancy) to permit error detection or correction at the receiver
Done to prevent the output of erroneous bits despite noise and other imperfections in the channel
The positions of the error control coding and decoding are shown in the transmission model

3. Transmission Model X(w) Hc(w) Transmitter N(w) + Y(w) Receiver Digital
Source
Encoder
Error
Control
Coding
Line
Modulator
(Transmit
Filter, etc)
Channel
Noise
Sink
Decoder
Decoding
Demod
(Receive +
Transmitter
Receiver
X(w)
Hc(w)
N(w)
Y(w)

4. Error Models Binary Symmetric Memoryless Channel
Assumes transmitted symbols are binary
Errors affect ‘0’s and ‘1’s with equal probability (i.e., symmetric)
Errors occur randomly and are independent from bit to bit (memoryless)
1-p
p is the probability of bit error or the Bit Error Rate (BER) of the channel p IN
OUT p 1 1
1-p

5. Error Models Many other types
Burst errors, i.e., contiguous bursts of bit errors
output from DFE (error propagation)
common in radio channels
Insertion, deletion and transposition errors
We will consider mainly random errors

6. Error Control Techniques
Error detection in a block of data
Can then request a retransmission, known as automatic repeat request (ARQ) for sensitive data
Appropriate for
Low delay channels
Channels with a return path
Not appropriate for delay sensitive data, e.g., real time speech and data

7. Error Control Techniques
Forward Error Correction (FEC)
Coding designed so that errors can be corrected at the receiver
Appropriate for delay sensitive and one-way transmission (e.g., broadcast TV) of data
Two main types, namely block codes and convolutional codes. We will only look at block codes

8. Block Codes We will consider only binary data
Data is grouped into blocks of length k bits (dataword)
Each dataword is coded into blocks of length n bits (codeword), where in general n>k
This is known as an (n,k) block code

9. Block Codes A vector notation is used for the datawords and codewords,
Dataword d = (d1 d2….dk)
Codeword c = (c1 c2……..cn)
The redundancy introduced by the code is quantified by the code rate,
Code rate = k/n
i.e., the higher the redundancy, the lower the code rate

10. Block Code - Example Dataword length k = 4 Codeword length n = 7
This is a (7,4) block code with code rate = 4/7
For example, d = (1101), c = ( )

11. Codeword + possible errors (n bits)
Error Control Process
Source code data chopped into blocks
Codeword (n bits)
101101
1000
Channel coder
1000
Dataword (k bits)
Codeword + possible errors (n bits)
Dataword (k bits)
Channel decoder
Channel
Error flags

12. Error Control Process Decoder gives corrected data
May also give error flags to
Indicate reliability of decoded data
Helps with schemes employing multiple layers of error correction

13. Parity Codes Example of a simple block code – Single Parity Check Code
In this case, n = k+1, i.e., the codeword is the dataword with one additional bit
For ‘even’ parity the additional bit is,
For ‘odd’ parity the additional bit is 1-q
That is, the additional bit ensures that there are an ‘even’ or ‘odd’ number of ‘1’s in the codeword

14. Parity Codes – Example 1 Even parity (i) d=(10110) so, c=(101101)

15. Parity Codes – Example 2 Coding table for (4,3) even parity code
Dataword
Codeword

16. Parity Codes To decode Code can detect single errors
Calculate sum of received bits in block (mod 2)
If sum is 0 (1) for even (odd) parity then the dataword is the first k bits of the received codeword
Otherwise error
Code can detect single errors
But cannot correct error since the error could be in any bit
For example, if the received dataword is (100000) the transmitted dataword could have been (000000) or (110000) with the error being in the first or second place respectively
Note error could also lie in other positions including the parity bit

17. Parity Codes
Known as a single error detecting code (SED). Only useful if probability of getting 2 errors is small since parity will become correct again
Used in serial communications
Low overhead but not very powerful
Decoder can be implemented efficiently using a tree of XOR gates

18. Hamming Distance
Error control capability is determined by the Hamming distance
The Hamming distance between two codewords is equal to the number of differences between them, e.g.,
have a Hamming distance = 3
Alternatively, can compute by adding codewords (mod 2)
= (now count up the ones)

19. Hamming Distance
The Hamming distance of a code is equal to the minimum Hamming distance between two codewords
If Hamming distance is:
1 – no error control capability; i.e., a single error in a received codeword yields another valid codeword
XXXXXXX X is a valid codeword
Note that this representation is diagrammatic only.
In reality each codeword is surrounded by n codewords. That is, one for every bit that could be changed

20. Hamming Distance If Hamming distance is:
2 – can detect single errors (SED); i.e., a single error will yield an invalid codeword
XOXOXO X is a valid codeword
O in not a valid codeword
See that 2 errors will yield a valid (but incorrect) codeword

21. Hamming Distance If Hamming distance is:
3 – can correct single errors (SEC) or can detect double errors (DED)
XOOXOOX X is a valid codeword
O in not a valid codeword
See that 3 errors will yield a valid but incorrect codeword

22. Hamming Distance - Example
Hamming distance 3 code, i.e., SEC/DED
Or can perform single error correction (SEC) X O O X
This code corrected this way
X is a valid codeword
O is an invalid codeword

23. Hamming Distance The maximum number of detectable errors is
That is the maximum number of correctable errors is given by,
where dmin is the minimum Hamming distance between 2 codewords and means the smallest integer

24. Linear Block Codes
As seen from the second Parity Code example, it is possible to use a table to hold all the codewords for a code and to look-up the appropriate codeword based on the supplied dataword
Alternatively, it is possible to create codewords by addition of other codewords. This has the advantage that there is now no longer the need to held every possible codeword in the table.

25. Linear Block Codes
If there are k data bits, all that is required is to hold k linearly independent codewords, i.e., a set of k codewords none of which can be produced by linear combinations of 2 or more codewords in the set.
The easiest way to find k linearly independent codewords is to choose those which have ‘1’ in just one of the first k positions and ‘0’ in the other k-1 of the first k positions.

26. Linear Block Codes
For example for a (7,4) code, only four codewords are required, e.g.,
So, to obtain the codeword for dataword 1011, the first, third and fourth codewords in the list are added together, giving
This process will now be described in more detail

27. Linear Block Codes An (n,k) block code has code vectors
d=(d1 d2….dk) and
c=(c1 c2……..cn)
The block coding process can be written as c=dG
where G is the Generator Matrix

28. Linear Block Codes Thus, ai must be linearly independent, i.e.,
Since codewords are given by summations of the ai vectors, then to avoid 2 datawords having the same codeword the ai vectors must be linearly independent

29. Linear Block Codes
Sum (mod 2) of any 2 codewords is also a codeword, i.e.,
Since for datawords d1 and d2 we have;
So,

30. Linear Block Codes 0 is always a codeword, i.e.,
Since all zeros is a dataword then,

31. Error Correcting Power of LBC
The Hamming distance of a linear block code (LBC) is simply the minimum Hamming weight (number of 1’s or equivalently the distance from the all 0 codeword) of the non-zero codewords
Note d(c1,c2) = w(c1+ c2) as shown previously
For an LBC, c1+ c2=c3
So min (d(c1,c2)) = min (w(c1+ c2)) = min (w(c3))
Therefore to find min Hamming distance just need to search among the 2k codewords to find the min Hamming weight – far simpler than doing a pair wise check for all possible codewords.

32. Linear Block Codes – example 1
For example a (4,2) code, suppose;
a1 = [1011]
a2 = [0101]
For d = [1 1], then;

33. Linear Block Codes – example 2
A (6,5) code with
Is an even single parity code

34. Systematic Codes
For a systematic block code the dataword appears unaltered in the codeword – usually at the start
The generator matrix has the structure, k R
R = n - k
P is often referred to as parity bits

35. Systematic Codes
I is k*k identity matrix. Ensures dataword appears as beginning of codeword
P is k*R matrix.

36. Decoding Linear Codes One possibility is a ROM look-up table
In this case received codeword is used as an address
Example – Even single parity check code;
Address Data
……… .
Data output is the error flag, i.e., 0 – codeword ok,
If no error, dataword is first k bits of codeword
For an error correcting code the ROM can also store datawords

37. Decoding Linear Codes
Another possibility is algebraic decoding, i.e., the error flag is computed from the received codeword (as in the case of simple parity codes)
How can this method be extended to more complex error detection and correction codes?

38. Parity Check Matrix
A linear block code is a linear subspace Ssub of all length n vectors (Space S)
Consider the subset Snull of all length n vectors in space S that are orthogonal to all length n vectors in Ssub
It can be shown that the dimensionality of Snull is n-k, where n is the dimensionality of S and k is the dimensionality of Ssub
It can also be shown that Snull is a valid subspace of S and consequently Ssub is also the null space of Snull

39. Parity Check Matrix
Snull can be represented by its basis vectors. In this case the generator basis vectors (or ‘generator matrix’ H) denote the generator matrix for Snull - of dimension n-k = R
This matrix is called the parity check matrix of the code defined by G, where G is obviously the generator matrix for Ssub- of dimension k
Note that the number of vectors in the basis defines the dimension of the subspace

40. Parity Check Matrix
So the dimension of H is n-k (= R) and all vectors in the null space are orthogonal to all the vectors of the code
Since the rows of H, namely the vectors bi are members of the null space they are orthogonal to any code vector
So a vector y is a codeword only if yHT=0
Note that a linear block code can be specified by either G or H

41. Parity Check Matrix So H is used to check if a codeword is valid,
R = n - k
The rows of H, namely, bi, are chosen to be orthogonal to rows of G, namely ai
Consequently the dot product of any valid codeword with any bi is zero

42. Parity Check Matrix This is so since, and so,
This means that a codeword is valid (but not necessarily correct) only if cHT = 0. To ensure this it is required that the rows of H are independent and are orthogonal to the rows of G
That is the bi span the remaining R (= n - k) dimensions of the codespace

43. Parity Check Matrix
For example consider a (3,2) code. In this case G has 2 rows, a1 and a2
Consequently all valid codewords sit in the subspace (in this case a plane) spanned by a1 and a2
In this example the H matrix has only one row, namely b1. This vector is orthogonal to the plane containing the rows of the G matrix, i.e., a1 and a2
Any received codeword which is not in the plane containing a1 and a2 (i.e., an invalid codeword) will thus have a component in the direction of b1 yielding a non- zero dot product between itself and b1

44. Parity Check Matrix
Similarly, any received codeword which is in the plane containing a1 and a2 (i.e., a valid codeword) will not have a component in the direction of b1 yielding a zero dot product between itself and b1 c1 c2 c3 a1 a2 b1

45. Error Syndrome
For error correcting codes we need a method to compute the required correction
To do this we use the Error Syndrome, s of a received codeword, cr
s = crHT
If cr is corrupted by the addition of an error vector, e, then
cr = c + e
and
s = (c + e) HT = cHT + eHT
s = 0 + eHT
Syndrome depends only on the error

46. Error Syndrome
That is, we can add the same error pattern to different codewords and get the same syndrome.
There are 2(n - k) syndromes but 2n error patterns
For example for a (3,2) code there are 2 syndromes and 8 error patterns
Clearly no error correction possible in this case
Another example. A (7,4) code has 8 syndromes and 128 error patterns.
With 8 syndromes we can provide a different value to indicate single errors in any of the 7 bit positions as well as the zero value to indicate no errors
Now need to determine which error pattern caused the syndrome

47. Error Syndrome
For systematic linear block codes, H is constructed as follows,
G = [ I | P] and so H = [-PT | I]
where I is the k*k identity for G and the R*R identity for H
Example, (7,4) code, dmin= 3

48. Error Syndrome - Example
For a correct received codeword cr = [ ]
In this case,

49. Error Syndrome - Example
For the same codeword, this time with an error in the first bit position, i.e.,
cr = [ ]
In this case a syndrome 001 indicates an error in bit 1 of the codeword

50. Comments about H
The minimum distance of the code is equal to the minimum number of columns (non-zero) of H which sum to zero
We can express
Where do, d1, dn-1 are the column vectors of H
Clearly crHT is a linear combination of the columns of H

51. Comments about H
For a codeword with weight w (i.e., w ones), then crHT is a linear combination of w columns of H.
Thus we have a one-to-one mapping between weight w codewords and linear combinations of w columns of H
Thus the min value of w is that which results in crHT=0, i.e., codeword cr will have a weight w (w ones) and so dmin = w

52. Comments about H
For the example code, a codeword with min weight (dmin = 3) is given by the first row of G, i.e., [ ]
Now form linear combination of first and last 2 cols in H, i.e., [011]+[010]+[001] = 0
So need min of 3 columns (= dmin) to get a zero value of cHT in this example

53. Standard Array
From the standard array we can find the most likely transmitted codeword given a particular received codeword without having to have a look-up table at the decoder containing all possible codewords in the standard array
Not surprisingly it makes use of syndromes

54. Standard Array The Standard Array is constructed as follows,
c1 (all zero) e1 e2 e3 … eN
c2+e1
c2+e2
c2+e3 ……
c2+eN c2
cM+e1
cM+e2
cM+e3
cM+eN cM s0 s1 s2 s3 sN
All patterns in row have same syndrome
Different rows have distinct syndromes
The array has 2k columns (i.e., equal to the number of valid codewords) and 2R rows (i.e., the number of syndromes)

55. Standard Array
The standard array is formed by initially choosing ei to be,
All 1 bit error patterns
All 2 bit error patterns ……
Ensure that each error pattern not already in the array has a new syndrome. Stop when all syndromes are used

56. Standard Array
Imagine that the received codeword (cr) is c2 + e3 (shown in bold in the standard array)
The most likely codeword is the one at the head of the column containing c2 + e3
The corresponding error pattern is the one at the beginning of the row containing c2 + e3
So in theory we could implement a look-up table (in a ROM) which could map all codewords in the array to the most likely codeword (i.e., the one at the head of the column containing the received codeword)
This could be quite a large table so a more simple way is to use syndromes

57. Standard Array This block diagram shows the proposed implementation
Compute syndrome
Look-up table + cr s e c

58. Standard Array
For the same received codeword c2 + e3, note that the unique syndrome is s3
This syndrome identifies e3 as the corresponding error pattern
So if we calculate the syndrome as described previously, i.e., s = crHT
All we need to do now is to have a relatively small table which associates s with their respective error patterns. In the example s3 will yield e3
Finally we subtract (or equivalently add in modulo 2 arithmetic) e3 from the received codeword (c2 + e3) to yield the most likely codeword, c2

59. Hamming Codes
We will consider a special class of SEC codes (i.e., Hamming distance = 3) where,
Number of parity bits R = n – k and n = 2R – 1
Syndrome has R bits
0 value implies zero errors
2R – 1 other syndrome values, i.e., one for each bit that might need to be corrected
This is achieved if each column of H is a different binary word – remember s = eHT

60. Hamming Codes Systematic form of (7,4) Hamming code is,
The original form is non-systematic,
Compared with the systematic code, the column orders of both G and H are swapped so that the columns of H are a binary count

61. Hamming Codes
The column order is now 7, 6, 1, 5, 2, 3, 4, i.e., col. 1 in the non-systematic H is col. 7 in the systematic H.

62. Hamming Codes - Example
For a non-systematic (7,4) code
d = 1011
c = =
e =
cr=
s = crHT = eHT = 011
Note the error syndrome is the binary address of the bit to be corrected

63. Hamming Codes
Double errors will always result in wrong bit being corrected, since
A double error is the sum of 2 single errors
The resulting syndrome will be the sum of the corresponding 2 single error syndromes
This syndrome will correspond with a third single bit error
Consequently the ‘corrected’ codeword will now contain 3 bit errors, i.e., the original double bit error plus the incorrectly corrected bit!

64. Bit Error Rates after Decoding
For a given channel bit error rate (BER), what is the BER after correction (assuming a memoryless channel, i.e., no burst errors)?
To do this we will compute the probability of receiving 0, 1, 2, 3, …. errors
And then compute their effect

65. Bit Error Rates after Decoding
Example – A (7,4) Hamming code with a channel BER of 1%, i.e., p = 0.01
P(0 errors received) = (1 – p)7 =
P(1 error received) = 7p(1 – p)6 =
P(3 or more errors) = 1 – P(0) – P(1) – P(2) =

66. Bit Error Rates after Decoding
Single errors are corrected, so,
= codewords are correctly detected
Double errors cause 3 bit errors in a 7 bit codeword, i.e., (3/7)*4 bit errors per 4 bit dataword, that is 3/7 bit errors per bit.
Therefore the double error contribution is 0.002*3/7 =

67. Bit Error Rates after Decoding
The contribution of triple or more errors will be less than (since the worst that can happen is that every databit becomes corrupted)
So the BER after decoding is approximately = = 0.09%
This is an improvement over the channel BER by a factor of about 11

68. Perfect Codes
If a codeword has n bits and we wish to correct up to t errors, how many parity bits (R) are needed?
Clearly we need sufficient error syndromes (2R of them) to identify all error patterns up to t errors
Need 1 syndrome to represent 0 errors
Need n syndromes to represent all 1 bit errors
Need n(n-1)/2 to syndromes to represent all 2 bit errors
Need nCe = n!/(n-e)!e! syndromes to represent all e bit errors

69. Perfect Codes
So,
If equality then code is Perfect
Only known perfect codes are SEC Hamming codes and TEC Golay (23,12) code (dmin=7). Using previous equation yields

70. Summary In this section we have
Used block codes to add redundancy to messages to control the effects of transmission errors
Encoded and decoded messages using Hamming codes
Determined overall bit error rates as a function of the error control strategy