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Hamming Code By Rituraj Kaushik & Mahendra Kumar Jajoo

Introduction

A Hamming code is a linear error detecting code.

It was developed by Richard Hamming.

An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers.IntroductionHamming codes can detect upto double bit errors and correct single bit error in a dataword.

It is more efficient than Parity code and Repetition code.

Coding and decoding algorithm Mark all bit positions that are powers of two as parity bits. (Positions 1, 2, 4, 8, 16, 32, 64, etc.)All other bit positions are for the data to be encoded. (positions 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, etc.)Eg: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .

Coding and decoding algorithm Each parity bit calculates the parity for some of the bits in the code word. The position of the parity bit determines the sequence of bits that it alternately checks and skips.Lets parity bits are denoted by p1, p2, p4, p8, p16, p32etc.

Coding and decoding algorithm P1 : check 1 bit, skip 1 bit, check 1 bit, skip 1 bit and so on.P2 : check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc. P4 : check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc.

Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits etc.

Coding and decoding algorithm

ExampleLets consider Hamming(11,7) code

Here no. of data bits = 7

Parity bits= 11-7 = 4

Total no. of bits = 11

ExampleLet the 7 bit data be 0110101

ExampleThe new data word (with parity bits) is now "10001100101". Lets assume that the LSB of the data gets corrupted and becomes "10001100100".

Now we apply reverse of the encoding algorithm to detect and correct the received data.

Example

Example So the error bit is the 11 th bit.

So corrected received data is 10001100101 The actual data is 0110101(Ignoring parity bits)

Hamming code with an extra parity Addition of an extra parity-checking bit to the end of the string which checks the entire string.

This will increase the error detection to 2 bit errors.

Applications They are widely used in computer memory (RAM).They are extremely common in the DRAM of computers used in satellites and space probes.Also used in NAND Flash memories to ensure data integrity.Hamming codes along with other codes are now-a-days being used in image processing, data hiding and data compression also.Advanced application Researches are going on applying Modified Hamming codes in optical channels with quantum noises and external noises. Modified Hamming codes will be able to exclude almost all errors on an optical channel.

Thank You

Thank you