ERROR CONTROL CODING

1
ERROR CONTROL CODING

• Basic concepts
• Classes of codes
• Block Codes
• Linear Codes
• Cyclic Codes
• Convolutional Codes

2
Basic Concepts

• Example Binary Repetition Codes
• (3,1) code 0 gt 000 1 gt 111
• Received 011. What was transmitted?
• scenario A 111 with one error in 1st location
• scenario B 000 with two errors in 2nd 3rd
  locations.
• Decoding
• P(A) (1- p)2 p
• P(B) (1- p) p2
• P(A) gt P(B) (for plt0.5)
• Decoding decision 011 gt 111

3
Probability of Error After Decoding

• (3,1) repetition code can correct single errors.
• In general for a tc-error correcting code
• Bit error probability for the (3,1)
  code, Pb Pu
• Gain For a BSC with p 10-2, Pb3x10-4.
• Cost Expansion in bandwidth or lower rate.

4
Hamming Distance

• Def. The Hamming distance between two codewords
  ci and cj, denoted by d(ci,cj), is the number of
  components at which they differ.
• dH(011,000) 2 dH
  C1,C2WH(C1C2)
• dH (011,111) 1
• Therefore 011 is closer to 111.
• Maximum Likelihood Decoding reduces to Minimum
  Distance Decoding, if the priory probabilities
  are equal (P(0)P(1))

5
Geometrical IllustrationHamming Cube
000
001
011
010
101
100
111
110
6
Error Correction and Detection

• Consider a code consisting of two codewords with
  Hamming distance dmin. How many errors can be
  detected? Corrected?
• of errors that can be detected td dmin -1
• of errors that can be corrected tc
• In other words, for t-error correction, we must
  have
• dmin 2tc 1

7
Error Correction and Detection (contd)

• Example dmin 5
• Can correct two errors
• Or, detect four errors
• Or, correct one error and detect two more errors.
• In general
• d min 2tc td 1

d min gt 2tc 1 d min gttc td 1
8
Minimum Distance of a Code

• Def. The minimum distance of a code C is the
  minimum Hamming distance between any two
  different codewords.
• A code with minimum distance dmin can correct all
  error patterns up to and including t-error
  patterns, where
• dmin 2tc 1
• It may be able to correct some higher error
  patterns, but not all.

9
Example (7,4) Code
10
Coding Gain and Cost (Revisited)

• Given an (n,k) code.
• Gain is proportional to the error correction
  capability, tc.
• Cost is proportional to the number of check
  digits, n-k r.
• Given a sequence of k information digits, it is
  desired to add as few check digits r as possible
  to correct as many errors (t) as possible.
• What is the relation between these code
  parameters?
• Note some text books uses m rather than r for the
  number check bits

11
Hamming Bound

• For an (n,k) code, there are 2k codewords and 2n
  possible received words.
• Think of the 2k codewords as centers of spheres
  in an n-dimensional space.
• All received words that differ from codeword ci
  in tc or less positions lie within the sphere Si
  of center ci and radius tc.
• For the code to be tc-error correcting (i.e. any
  tc-error pattern for any codeword transmitted can
  be corrected), all spheres Si , i 1,.., 2k ,
  must be non-overlapping.

12
Hamming Bound (contd)

• In other words, When a codeword is selected, none
  of the n-bit sequences that differ from that
  codeword by tc or less locations can be selected
  as a codeword.
• Consider the all-zero codeword. The number of
  words that differ from this codeword by j
  locations is
• The total number of words in any sphere
  (including the codeword at the center) is

13
Hamming Bound (contd)

• The total number of n-bit sequences that must be
  available (for the code to be a tc-error
  correcting code) is
• But the total number of sequences is 2n.
  Therefore

14
Hamming Bound (contd)

• The above bound is known as the Hamming Bound. It
  provides a necessary, but not a sufficient,
  condition for the construction of an (n,k)
  tc-error correcting code.
• Example Is it theoretically possible to design a
  (10,7) single-error correcting code?
• A code for which the equality is satisfied is
  called a perfect code.
• There are only three types of perfect codes
  (binary repetition codes, the hamming codes, and
  the Golay codes).
• Perfect does not mean best!

15
Gilbert Bound

• While Hamming bound sets a lower limit on the
  number of redundant bits (n-k) required to
  correct tc errors in an (n,k) linear block code.
• Another lower limit is the Singleton bound
• Gilbert bound places an upper bound on the number
  of redundant bits required to correct tc errors.
• It only says there exist a code but it does not
  tell you how to find it.

16
The Encoding Problem

• How to select 2k codewords of the code C from the
  2n sequences such that some specified (or
  possibly the maximum possible) minimum distance
  of the code is guaranteed?
• Example How were the 16 codewords of the (7,4)
  code constructed? Exhaustive search is
  impossible, except for very short codes (small k
  and n)
• Are we going to store the whole table of 2k(nk)
  entries?!
• A constructive procedure for encoding is
  necessary.

17
The Decoding Problem

• Standard Array
• 0000000 1101000 0110100 1011100 1110010
  0011010 1000110 0101110 1010001 0111001
  1100101 0001101 0100011 1001011 0010111
  1111111
• 0000001 1101001 0110101 1011101 1110011
  0011011 1000111 0101111 1010000 0111000
  1100100 0001100 0100010 1001010 0010110
  1111110
• 0000010 1101010 0110110 1011110 1110000
  0011000 1000100 0101100 1010011 0111011
  1100111 0001111 0100001 1001001 0010101
  1111101
• 0000100 1101100 0110000 1011000 1110110
  0011110 1000010 0101010 1010101 0111101
  1100001 0001001 0100111 1001111 0010011
  1111011
• 0001000 1100000 0111100 1010100 1111010
  0010010 1001110 0100110 1011001 010001
  1101101 0000101 0101011 1000011 0011111
  1110111
• 0010000 1111000 0100100 1001100 1100010
  0001010 1010110 0111110 1000001 0101001
  1110101 0011101 0110011 1011011 0000111
  1101111
• 0100000 1001000 0010100 1111100 1010010
  0111010 1100110 0001110 1110001 0011001
  1000101 0101101 0000011 1101011 0110111
  1011111
• 1000000 0101000 1110100 0011100 0110010
  1011010 0000110 1101110 0010001 1111001
  0100101 1001101 1100011 0001011 1010111
  0111111
• Exhaustive decoding is impossible!!
• Well-constructed decoding methods are required.
• Two possible types of decoders
• Complete always chooses minimum distance
• Bounded-distance chooses the minimum distance up
  to a certain tc. Error detection is utilized
  otherwise.