1. ERROR DETECTING AND CORRECTING CODES -BY R.W. HAMMING PRESENTED BY- BALAKRISHNA DHARMANA

2. INTRODUCTION Why do we need error detection and correction? Unwanted Random signals interfere with accurate transmission of signals Some simple ways of error detection and correction Sending each word again Sending each letter again Within a computer errors are rare

3. Systematic codes Redundancy R= n/m Redundancy serves to measure the efficiency of the code Lowers the effective channel capacity

4. TYPES OF CODES Single error detecting codes Single error correcting codes Single error correcting plus double error detecting codes

5. Application of these codes may be expected to occur under conditions:- Unattended operation over long periods of time Extremely large and tightly interrelated systems where a single failure causes the entire installation When the signaling is not possible in the presence of noise

6. Contains n-bits Out of n-bits, n-1 are information bits and one parity bit Redundancy = n/n-1 As n increases probability of getting errors increases Type of check used to detect any single error is called parity check (even or odd) SINGLE ERROR DETECTING CODES

7. SINGLE ERROR CORRECTING CODES First assign m positions in available positions as information positions Specific positions are left to a later determination Assign k remaining positions as check positions Apply k parity checks

8. The result of the k parity checks from right to left is checking number Checking number must describe m+k+1 different things so that, 2 k >= m + k + 1 writing n = m+k, we find 2 m <= 2 n / n+1

10. Now we have to determine the positions over which the various parity checks are to be applied Any position which has a 1 on the right of it’s binary representation must cause the first check fail. By examining the binary form of the various integers 1 - 1 3 - 11 5 - 101 7 - 111 etc

11. Check number 1 2 3 4. Check positions 1 2 4 8. Positions checked 1,3,5,7,9,11,…………… 2,3,6,7,10,11,…………. 4,5,6,7,12,13,…………. 8,9,10,11,12,13,………. TABLE II

13. SINGLE ERROR CORRECTING PLUS DOUBLE ERROR DETECTING CODES Begin with single error correcting code Add one more position for checking all previous positions using even parity check In the operation of the code, No errors – all parity checks including the last are satisfied Single error- the last parity check fails Two errors- last parity check is satisfied and indicates some kind of error

14. GEOMETRICAL MODEL

15. Minimum dist 1 2 3 4 5 meaning Uniqueness Single error detection Single error correction Single error correction plus double error detection Double error correction

16. At a given minimum distance, some of the correctability can be exchanged for more detectability. For example, a subset with minimum distance 5 may be used for: Double error correction Single error correction plus triple error detection Quadruple error detection

17. If code points are at a distance of at least 2 from each other then – any single error will carry the code point over to a point that is not a code point. Means – single error is detectable If distance is at least 3 units then any single error will leave the point nearer to the correct code point than to any other code point, this means – single error will be correctable. APPLICATION OF GEOMETRICAL MODEL TO CODES

18. CONCLUSION This paper helps us to discuss the minimum redundancy code techniques for Single error detection Single error correction And single error correction plus double error detection Also gives the geometrical model of above techniques in depth.

19. REFERENCE M. J. E. Golay, Correspondence, notes on Digital coding, Proceedings of the I.R.E., Vol. 37, p. 657, June 1949. http://www.math.ups.edu/~bryans/current/journal_s pring_2002/300_EFejta_2002.htm http://www.math.ups.edu/~bryans/current/journal_s pring_2002/300_EFejta_2002.htm http://www.ee.unb.ca/tervo/ee4253/hamming.htm http://www.cs.mdx.ac.uk/staffpages/mattsmith/mod ules/COM1021/seminar_sheets http://www.cs.mdx.ac.uk/staffpages/mattsmith/mod ules/COM1021/seminar_sheets