it.information theory – How does the choice of basis of parity check matrix affect the tanner graph

Given a binary linear block code defined by its parity check matrix $H$, many important characteristics of this code are determined by graph theoretic properties of the Tanner graph $Gamma$ derived from $H$. The girth of $Gamma$ and the distribution of low length cycles greatly affect the BER (bit error rate) performance when the commonly used iterative decoding algorithms are used (belief-propagation,…). We’re free to choose linear combinations of the rows of $H$ as another basis; this doesn’t affect the code (same codewords,…), but how does that affect the Tanner graph and its properties? What (if anything) is invariant under a change of basis?