Which Codes Have$4$-Cycle-Free Tanner Graphs?

Let C be an [n,k,d] binary linear code with rate R=k/n and dual C
^perp
. In this correspondence, it is shown that C can be represented by a 4-cycle-free Tanner graph only if: pd
^perp
les lfloorradicnp(p-1)+n
²
/4+n/2 rfloorwhere p=n-k and d
^perp
is the minimum distance of C
^perp
. By applying this result, it is shown that 4-cycle- free Tanner graphs do not exist for many classical binary linear block codes