Editing Block code (section)

== Error detection and correction properties ==

A codeword <math>c \in \Sigma^n</math>could be considered as a point in the <math>n</math>-dimension space <math>\Sigma^n</math> and the code <math>\mathcal{C}</math> is the subset of <math>\Sigma^n</math>. A code <math>\mathcal{C}</math> has distance <math>d</math> means that <math>\forall c\in \mathcal{C}</math>, there is no other codeword in the [[Hamming ball]] centered at <math>c</math> with radius <math>d-1</math>, which is defined as the collection of <math>n</math>-dimension words whose ''[[Hamming distance]]'' to <math>c</math> is no more than <math>d-1</math>. Similarly, <math> \mathcal{C}</math> with (minimum) distance <math>d</math> has the following properties:
* <math> \mathcal{C}</math> can detect <math>d-1</math> errors : Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, no error pattern of <math>d-1</math> or fewer errors could change one codeword to another. When the receiver detects that the received vector is not a codeword of <math> \mathcal{C}</math>, the errors are detected (but no guarantee to correct).
* <math> \mathcal{C}</math> can correct <math>\textstyle\left\lfloor {{d-1} \over 2}\right\rfloor</math> errors. Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, the two Hamming balls centered at two different codewords respectively with both radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math> do not overlap with each other. Therefore, if we consider the error correction as finding the codeword closest to the received word <math>y</math>, as long as the number of errors is no more than <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, there is only one codeword in the hamming ball centered at <math>y</math> with radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, therefore all errors could be corrected.
* In order to decode in the presence of more than <math>(d-1)/2</math> errors, [[list-decoding]] or [[Decoding methods#Maximum likelihood decoding|maximum likelihood decoding]] can be used.
* <math> \mathcal{C}</math> can correct <math>d-1</math> [[Binary erasure channel|erasures]]. By ''erasure'' it means that the position of the erased symbol is known. Correcting could be achieved by <math>q</math>-passing decoding : In <math>i^{th}</math> passing the erased position is filled with the <math>i^{th}</math> symbol and error correcting is carried out. There must be one passing that the number of errors is no more than <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math> and therefore the erasures could be corrected.