Editing Block code (section)

=== {{anchor|Minimum distance}}The distance ''d'' ===
The '''distance''' or '''minimum distance''' {{mvar|d}} of a block code is the minimum number of positions in which any two distinct codewords differ, and the '''relative distance''' <math>\delta</math> is the fraction <math>d/n</math>.
Formally, for received words <math>c_1,c_2\in\Sigma^n</math>, let <math>\Delta(c_1,c_2)</math> denote the [[Hamming distance]] between <math>c_1</math> and <math>c_2</math>, that is, the number of positions in which <math>c_1</math> and <math>c_2</math> differ.
Then the minimum distance <math>d</math> of the code <math>C</math> is defined as
:<math>d := \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)]</math>.
Since any code has to be [[injective]], any two codewords will disagree in at least one position, so the distance of any code is at least <math>1</math>. Besides, the '''distance''' equals the '''[[Hamming weight#Minimum weight|minimum weight]]''' for linear block codes because:{{cn|date=December 2024}}
:<math>\min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)] = \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[\mathbf{0},C(m_2)-C(m_1)] = \min_{m\in\Sigma^k\atop m\neq\mathbf{0}} w[C(m)] = w_\min</math>.

A larger distance allows for more error correction and detection.
For example, if we only consider errors that may change symbols of the sent codeword but never erase or add them, then the number of errors is the number of positions in which the sent codeword and the received word differ.
A code with distance {{mvar|d}} allows the receiver to detect up to <math>d-1</math> transmission errors since changing <math>d-1</math> positions of a codeword can never accidentally yield another codeword. Furthermore, if no more than <math>(d-1)/2</math> transmission errors occur, the receiver can uniquely decode the received word to a codeword. This is because every received word has at most one codeword at distance <math>(d-1)/2</math>. If more than <math>(d-1)/2</math> transmission errors occur, the receiver cannot uniquely decode the received word in general as there might be several possible codewords. One way for the receiver to cope with this situation is to use [[list decoding]], in which the decoder outputs a list of all codewords in a certain radius.