error detection & correction by illuminati

ERROR DETECTION &

CORRECTION

By: ILLUMINATI

Group Members

Anil Mohanty Mohammad Ali Dipesh Badlani Munaff Ahmad Khan

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Contents

1) Introduction2) Cyclic codes3) Cyclic redundancy check4) Hamming Distance5) Implementation6) Code7) Demo8) Acknowledgements

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Introduction

The aim of an error detection technique is to enable the receiver of a message transmitted through noisy channel to determine whether the message has been corrupted.

To do this, the transmitter constructs a value called a checksum that is the function of message and appends it to message.

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The receiver can then use the same function to calculate the checksum of the received message and compare it with the appended checksum to see if the message was correctly received.

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Cyclic Codes

Cyclic codes are special linear block codes with one extra property. In a cyclic code, if a codeword is cyclically shifted (rotated), the result is another codeword.

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Cyclic ReDUNDANCY CHECK (CRC)

A cyclic redundancy check (CRC) is a non-secure hash function designed to detect accidental changes to raw computer data, and is commonly used in digital networks and storage devices such as hard disk devices.

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CRCs are so called because the check (data verification) code is a redundancy (it adds zero information) and the algorithm is based on cyclic codes.

The term CRC may refer to the check code or to the function that calculates it, which accepts data streams of any length as input but always outputs a fixed-length code

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Generator Polynomial

The divisor in a cyclic code is normally called the generator polynomial or simply the generator.

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pOLynomial generator characteristics

1. It should have at least two terms.2. The coefficient of the term x0 should be 1.3. It should not divide xt + 1, for t between 2 and n − 1.4. It should have the factor x + 1.

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Commonly used CRC generator polynomials:

CRC C(x)

CRC-8 x^8+x^2+x^1+1

CRC-10 x^10+x^9+x^5+x^4+x^1+1

CRC-12 x^12+x^11+x^3+x^2+x^1+1

CRC-16 x ^16+x^15 +x ^2+1

CRC-CCIT X^16+x^12+x^5+1

CRC-32 X^32+x^26+x^23+x^22+x^16+x^12+x^11+x^10+x^8+x^7+x^5+x^4+x^2+x+1

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Hamming Distance (HD):

Hamming Distance (HD):HD between two code-words := number of bit positions in which two code-words

differHD of complete code: HD := min { HD of any two code-words}

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Required Hamming Distance for the two forms of error control:

To detect k errors, HD of complete code must be k+1.

To correct k errors, HD of complete code must be 2k+1.

In order to achieve the necessary Hamming Distance we usually have to add redundant bits

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A CRC code with C(7, 4)

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CRC encoder and decoder

Encoding Procedure At Sender

Multiply i(x) by x(n-k) by putting zeros in n-k low order positions

Divide x(n-k)i(x) by g(x) to get r(x). x(n-k)i(x) = g(x) q(x) + r(x) where q(x) is quotient and

r(x) is remainder  

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Division in CRC encoder

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Add remainder r(x) to x(n-k)i(x) by putting check bits in n-k lower order positions

Based on randomness, the message can be transmitted with error or without error.

For transmission with error, introduce an error at random position to the message x(n-k)i(x) and display the position of the error.

Transmitted codeword is b(x) = x(n-k)i(x) + r(x)

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 Decoding Procedure at the Receiver

The received message b(x) is divided by g(x) using

Euclidean division algorithm

If the remainder is 0 then there is no error in the

transmission else Error in the transmission.

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Division in the CRC decoder for two cases

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Using Polynomials We can use a polynomial to represent a

binary word. Each bit from right to left is mapped onto

a power term. The rightmost bit represents the “0”

power term. The bit next to it the “1” power term, etc.

If the bit is of value zero, the power term is deleted from the expression.

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A polynomial to represent a binary word

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CRC division using polynomials

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Acknowledgments

Thank you to the faculty and all of you present here.

A special thanks to the GROUP CHAAR MUSAAFIR without whom this wouldn’t have been possible.