Block code: Difference between revisions

Revision as of 05:01, 22 November 2022

Revision as of 05:01, 22 November 2022

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A large rate means that the amount of actual message per transmitted block is high. In this sense, the rate measures the transmission speed and the quantity <math>1-R</math> measures the overhead that occurs due to the encoding with the block code.

It is a simple [[information theory|information theoretical]] fact that the rate cannot exceed <math>1</math> since data cannot in general be losslessly compressed. Formally, this follows from the fact that the code <math>C</math> is an injective map.

=== {{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:

:<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_1)+C(m_2)] = \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.

=== Popular notation ===