Hamming Code

Rachel Ah Chuen

Basic concepts• Networks must be able to transfer data from one device

to another with complete accuracy.

• Data can be corrupted during transmission.

• For reliable communication, errors must be detected and corrected.

Types of Errors

Single-bit error

Single bit errors are the least likely type of errors in serial data transmission because the noise must have a very short duration which is very rare.

However this kind of errors can happen in parallel transmission.

Single-bit error

Burst error

The term burst error means that two or more bits in the data unit have changed from 1 to 0 or from 0 to 1.

Burst errors does not necessarily mean that the errors occur in consecutive bits.

The length of the burst is measured from the first corrupted bit to the last corrupted bit.

Burst error

Hamming Code In the late 1940’s Richard Hamming recognized

that the further evolution of computers required greater reliability, in particular the ability to not only detect errors, but correct them.

His search for error-correcting codes led to the Hamming Codes, and the extended Hamming Codes, 1-error correcting and 2-error detecting codes.

DefinitionsFor linear codes, encoding is a linear

transformation c that maps a message m to a codeword c(m).

Block codes means all codewords c(m) have the same length, which we denote as n.

Let k be the length of information bits that we encode. Then there are n-k redundancy bits in each codeword, called parity check bits, which are left for error correction.

(n, k) specify the parameters (codeword length and parity check bit length) for a block code.

Hamming DistanceThe Hamming distance, d_H is the number of bits

that have to be changed to get from one bit pattern to another.

Example: c1=(10010101) & c2=(10011001) have a hamming distance of 2

What about c1 = (101101) and c2 = (100110)?

Minimum Hamming DistanceLet d to be the minimum Hamming distance

between any two distinct codewords of a code C as

A code with minimum distance d between codewords can detect d-1 errors and can correct (d-1)/2 errors.

Hamming weightw(S) defined as the sum of its non-zero entries

A linear code is completely defined by all the codewords for messages of weight 1.

Hamming CodeThe Hamming code is a single error correction

linear block code with (n, k) = (2m -1, 2m-1-m)

where m = n k is the number of check bits in the codeword.

The simplest non-trivial code is for m = 3, which is the (7,4) Hamming code.

Generator matrix

Where S is k x (n k)

Hamming codeSuppose our matrix G generates a

single error correcting code. Then there are no code words of weight 1 or 2, and the parity check portion S of G must satisfy the following 2 conditions:

(7,4) Hamming Code

(7,4) Hamming CodeFor every generator matrix g=[ I S], there is a

parity check matrix H defined by

Examplereceived string r = (1110111)

Parity ErrorsThe class of Hamming single error correcting

codes is noteworthy because they are particularly efficient in the use of parity bits.

The (7,4) Hamming code uses 3 parity bits to protect 4 data bits;

(15,11) code

Parity Computation(7,4) code

Computations by receiver

Steps

Combination of syndrome bitsAppropriate corrective action for each

combination of the syndrome bits