Error Detection and Correction

Data may be modified so that errors can either be:

• detected (i.e. you can only tell that the data is wrong), or
• corrected (i.e. you can unambiguously tell what the data should have been, and hence you can confidentially correct it).

Correction is required where communication must be simplex (only possible in one direction), but correction is expensive. A good example of its need is between Earth and inter-planetary spacecraft. Error correction by the receiver is referred to as forward error correction, whereas re-transmission schemes are referred to as reverse error correction.

Codewords are constructed consisting of both data and check bits.

The Hamming distance between two codewords consists of the number of bit positions in which they differ.
The difference is performed Modulo-2 or EXCLUSIVE-OR:

e.g.	1 ⊕ 1  =  0 ⊕ 0  =  0
	1 ⊕ 0  =  0 ⊕ 1  =  1

	 10011101
	 10111110
     --------------
       ⊕ 00100011

Here, the Hamming distance is 3.

The Hamming distance of a code is the minimum Hamming distance between any two words in that code.