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 ==> ==> 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) > P(B) (for p<0.5) Decoding decision: 011 ==> 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, Pb=3x10-4. 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) = dH [C1,C2]=WH(C1+C2)
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 Illustration Hamming Cube
000
001
011
010
101
100
110
111

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 (cont’d)
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 > 2tc + 1
d min >tc + 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 (cont’d)
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 (cont’d)
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 (cont’d)
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(n+k) entries?!
A constructive procedure for encoding is necessary.

17. The Decoding Problem Standard Array `
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.