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

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2 Author(s)
Hagiwara, M. ; Res. Center for Inf. Security, Nat. Inst. of Adv. Ind. Sci. & Technol., Tokyo ; Fossorier, M.

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 e<sub>1</sub> = o<sub>2</sub> directly follows from o<sub>1</sub> + e<sub>1</sub> and o<sub>2</sub> + e<sub>2</sub> = e. The other equality o<sub>1</sub> = e<sub>2</sub> follows from e<sub>1</sub> = e<sub>2</sub> 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<sub>1,2</sub> (I) mod p, 0 < I < L - 1} = {0,1,..., L - 1} from [M. Fossorier, 2004, Theorem 2.1], so that Sigma<sub>I=0</sub><sup>L-1</sup> Delta<sub>1,2</sub> (I) = m mod p ne 0. Since Ruwei Chen recently pointed out this issue, we decided to clarify this point.

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Information Theory, IEEE Transactions on  (Volume:55 ,  Issue: 3 )