Index vector-function and minimal cycles

Let $P$ be a closed triangulated manifold, $\dim{P}=n$. We consider the group of simplicial 1-chains $C_1(P)=C_1(P,\mathbb Z_2)$ and the homology group $H_1(P)=H_1(P,\mathbb Z_2)$. We also use some nonnegative weighting function $L\colon C_1(P)\to\mathbb R$. For any homological class $[x]\in H_1(P)$ the method proposed in article builds a cycle $z\in[x]$ with minimal weight $L(z)$. The main idea is in using a simplicial scheme of space of the regular covering $p\colon\hat P\to P$ with automorphism group $G\cong H_1(P)$. We construct this covering applying the index vector-function $J\colon C_1(P)\to\mathbb Z_2^r$ relative to any basis of group $H_{n-1}(P)$, $r=\operatorname{rank}H_{n-1}(P)$.