= Parity-check dependencies determine minimum distance
{title2=$d(C)=\min\{|S|:\text{the coordinate checks indexed by }S\text{ are dependent}\}$}
In the usual column convention, a <codeword> $c$ of a <linear code> obeys $Hc^T=0$, so its nonzero coordinates give a <linear dependence> among columns of its <parity-check matrix>. Conversely a dependence gives a nonzero codeword supported on those coordinates. Minimizing the support gives the <minimum Hamming distance of a linear code>. In the transposed convention $cP=0$, the corresponding objects are the rows of $P$. The size of a dependent set means the number of its coordinate positions, even when some check vectors coincide.
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