Parity-check dependencies determine minimum distance ($d(C)=\min \{|S|:\text{the coordinate checks indexed by }S\text{ are dependent}\}$)

In the usual column convention, a codeword $c$ of a linear code obeys $H{c}^T=0$, 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.