In this paper, we present a new exact algorithm for counting perfect matchings, which relies on neither inclusion-exclusion principle nor tree-decompositions. For any bipartite graph of $2n$ nodes and $Δn$ edges such that $Δ\geq 3$, our algorithm runs with $O^{\ast}(2^{(1 - 1/O(Δ\log Δ))n})$ time and exponential space. Compared to the previous algorithms, it achieves a better time bound in the sense that the performance degradation to the increase of $Δ$ is quite slower. The main idea of our algorithm is a new reduction to the problem of computing the cut-weight distribution of the input graph. The primary ingredient of this reduction is MacWilliams Identity derived from elementary coding theory. The whole of our algorithm is designed by combining that reduction with a non-trivial fast algorithm computing the cut-weight distribution. To the best of our knowledge, the approach posed in this paper is new and may be of independent interest.