ERROR DETECTING AND ERROR DETECTING AND CORRECTING CODESCORRECTING CODES

-BY R.W. HAMMING

PRESENTED BY-

BALAKRISHNA DHARMANA

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

Systematic codes Redundancy

R= n/m Redundancy serves to measure the

efficiency of the code Lowers the effective channel capacity

TYPES OF CODES

Single error detecting codes

Single error correcting codes

Single error correcting plus double error detecting codes

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

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

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

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, 2k >= m + k + 1 writing n = m+k, we find 2m <= 2n / n+1

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

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

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

GEOMETRICAL MODEL

Minimum dist

1 2 3 4 5

meaning

UniquenessSingle error detectionSingle error correctionSingle error correction plus double error detectionDouble error correction

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

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

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.

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_spring_2002/300_EFejta_2002.htm

http://www.ee.unb.ca/tervo/ee4253/hamming.htm

http://www.cs.mdx.ac.uk/staffpages/mattsmith/modules/COM1021/seminar_sheets