Block code: Difference between revisions

Revision as of 00:28, 14 November 2024

Revision as of 20:19, 7 December 2024

Line 46:

:<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_1m_2)+-C(m_2m_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.