Generalizing a graph-theoretical result of Maffray to binary matroids, Oxley and Wetzler proved that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is isomorphic to $M(K_4)$ or $F_7$, or M is the cycle matroid of a graph consisting of a collection of triangles all sharing a common edge. In this paper, we show that if M is a 3-connected binary matroid having a 5-element circuit but no larger odd circuit, then M has rank less than six; or M has rank six and is one of nine sporadic matroids; or M can be obtained by attaching together, via generalized parallel connection across a common triangle, a collection of copies of $F_7$ and $M(K_4)$ and then possibly deleting up to two elements of the common triangle. From this, we deduce that a 3-connected simple graph with a 5-cycle but no larger odd cycle is obtained from $K_{3,n}$ for some $n\ge 3$ by adding one, two, or three edges between the vertices in the 3-vertex class.

将 Maffray 的图论结果推广到二元拟阵，Oxley 和 Wetzler 证明了连接的简单二元拟阵M除了三角形之外没有奇数回路当且仅当M是仿射的，M同构为$米({钾}_4)$ 或者 $F_7$，或M是由共享公共边的三角形集合组成的图的循环拟阵。在本文中，我们证明如果M是一个具有 5 元素电路但没有更大奇电路的 3 连通二进制拟阵，则M 的秩小于 6；或M有六阶并且是九个零星拟阵之一；或M可以通过在一个公共三角形上通过广义平行连接将一组副本连接在一起来获得$F_7$ 和 $米({钾}_4)$然后可能最多删除公共三角形的两个元素。由此，我们推导出一个具有 5 个圈但没有更大奇数圈的 3-连通简单图是从${钾}_{3,n}$ 对于一些 $n\ge 3$ 通过在 3-vertex 类中的顶点之间添加一条、两条或三条边。