The problem is NP-hard (even in the unweighted case) via a well-known connection to coding theory. Namely, if $A$ is the parity check matrix of a binary linear code $C$, then the distance of $C$ is the size of a shortest circuit in the binary matroid represented by $A$. In The intractability of computing the minimum distance of a code, Vardy proved that computing the distance of a binary linear code is NP-hard.

However, for proper minor-closed classes of matroids, Geelen, Gerards, and Whittle conjecture that the problem is solvable in polynomial-time. See the paper Computing girth and cogirth in perturbed graphic matroids or Finding Shortest Circuits in Binary Matroids on the Matroid Union Blog for more information.

The problem is NP-hard (even in the unweighted case) via a well-known connection to coding theory. Namely, if $A$ is the parity check matrix of a binary linear code $C$, then the distance of $C$ is the size of a shortest circuit in the binary matroid represented by $A$. In The intractability of computing the minimum distance of a code, Vardy proved that computing the distance of a binary linear code is NP-hard.

However, for proper minor-closed classes of binary matroids, Geelen, Gerards, and Whittle conjecture that the problem is solvable in polynomial-time. See the paper Computing girth and cogirth in perturbed graphic matroids or Finding Shortest Circuits in Binary Matroids on the Matroid Union Blog for more information.

The problem is NP-hard (even in the unweighted case) via a well-known connection to coding theory. Namely, if $A$ is the parity check matrix of a binary linear code $C$, then the distance of $C$ is the size of a shortest circuit in the binary matroid represented by $A$. In The intractability of computing the minimum distance of a code, Vardy proved that computing the distance of a binary linear code is NP-hard.

However, for proper minor-closed classes of matroids, Geelen, Gerards, and Whittle conjecture that the problem is solvable in polynomial-time. See Computing girth and cogirth in perturbed graphic matroids for more information.

The problem is NP-hard (even in the unweighted case) via a well-known connection to coding theory. Namely, if $A$ is the parity check matrix of a binary linear code $C$, then the distance of $C$ is the size of a shortest circuit in the binary matroid represented by $A$. In The intractability of computing the minimum distance of a code, Vardy proved that computing the distance of a binary linear code is NP-hard.

However, for proper minor-closed classes of matroids, Geelen, Gerards, and Whittle conjecture that the problem is solvable in polynomial-time. See the paper Computing girth and cogirth in perturbed graphic matroids or Finding Shortest Circuits in Binary Matroids on the Matroid Union Blog for more information.