Comment on "Quasi-Cyclic Low Density Parity Check Codes From Circulant Permutation Matrices"

While preparing [H. Hagiwara et al., 2006], we realized that the proof of [M. Fossorier, 2004, Theorem 2.3] was leading to confusion as written. More precisely, only e₁ = o₂ directly follows from o₁ + e₁ and o₂ + e₂ = e. The other equality o₁ = e₂ follows from e₁ = e₂ and the fact that the sum of the (distinct) Delta´s between the two rows considered has to be zero. Actually, a much concise proof can be obtained by directly observing that for J = p = 2m, {Delta_1,2 (I) mod p, 0 < I < L - 1} = {0,1,..., L - 1} from [M. Fossorier, 2004, Theorem 2.1], so that Sigma_I=0^L-1 Delta_1,2 (I) = m mod p ne 0. Since Ruwei Chen recently pointed out this issue, we decided to clarify this point.