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)+n²/4+n/2rfloor where 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