The existence of even cycles with specific lengths in Wenger's graph

Wenger's graph H-m(q) is a q-regular bipartite graph of order 2q(m) constructed by using the m-dimensional vector space F-q(m) over the finite field F-q. The existence of the cycles of certain even length plays an important role in the study of the accurate order of the Turan number ex(n; C-2m) in extremal graph theory. In this paper, we use the algebraic methods of linear system of equations over the finite field and the "critical zero-sum sequences" to show that: if m >= 3, then for any integer l with l not equal 5,4 <= l <= 2ch(F-q) (where ch(F-q) is the character of the finite field F-q) and any vertex v in the Wenger's graph H-m (q), there is a cycle of length 21 in H-m (q) passing through the vertex v.