我們提供了一個簡單的方法，憑藉著一些充份必要條件去設計具有高值最短周長，基於二次同餘方程式架構化的半循環式低密度同位元檢查碼。假如半循環式低密度同位元檢查碼的查核矩陣是由循環式排列矩陣所組成的，則這個方法是相當有效率的。並且，因為在同位元檢查碼中循環架構是由循環式排列矩陣裡的位移值所決定的，所以去發現沒有低值最短周長的位移值是很重要的。然而，具有高值最短周長的位移值是可以被”Fossorier”推演的充份必要條件所控制的，因此我們利用”Fossorier”定理去推導形成循環長度為6, 8, 10的方程式。假如某些齊次線性方式含有無解便可以使這個碼擁有高值的最短周長，換句話說，我們的方法消除了造成循環長度為6, 8, 10的因素。因此，這種具有高值最短周長的碼是有助於用電腦搜尋策略架構，並且直到它的條件不被滿足為止。事實上，這個技術可以被用來完全消除低值的循環長度，以致於碼的最短周長可以高逹12。模擬結果也證明了使用這個技術可以消除低值的循環，並導致在可加性高斯白雜訊通道裡有著明顯的訊號雜訊比增益。

We provide a simple method to design structured quasi-cyclic low-density parity-check (QC-LDPC) codes with large girth based on quadratic congruential equation via some simple necessary and sufficient condition. This approach is relatively effective if the parity-check matrix of a QC-LDPC code is composed of circulant permutation matrices, such as quadratic congruent code (QCC). Moreover, since the cycle structures in QC-LDPC codes are determined by shift values of circulant permutation matrices, it is important to find the proper shift values which make no short cycles. However, shift values with large girth can be controlled by some simple necessary and sufficient conditions which Fossorier derive, hence we utilize Fossorier’s theorem to derive the desired equations which form some cycles of length 6, 8, 10. If certain homogeneous linear equations contain no solution such that the codes can have large girth, in other words, our approach eliminates the factor that cause cycles of length 6, 8, and 10. Subsequently, such codes with large girth were constructed with the aid of computer search strategies until the conditions are not met. In particular, this technique can be used to entirely eliminate cycles of short lengths, resulting in the codes of girth up to twelve. Simulation results show that eliminating short cycles using this technique leads to significant signal-to-noise rate (SNR) gain, over the addition white Gaussian noise (AWGN) channel.

[1] R. G. Gallager, “Low-Density Parity-Check Codes,” Cambridge, MA: M.I.T, press. 1963.
[2] D. J. C. Mackey and R. M. Neal, “Near Shannon limit performance of low-density parity-check codes,” Electron. lett., vol. 32, pp. 1645-1646, 1996.
[3] Y. Kou, S. Lin, and M. fossorier, “low-density parity-check codes based on finite geometries: A rediscovery and new results,” IEEE. Trans. Inform. Theory, vol. 47, pp. 2711-2736, Nov. 2001.
[4] R. M. Tanner, D. Sridhara, and T. Fuja, “A class of group-structured LDPC codes,” in Proc. 6th Int. Symp. Commun. Theory and Applications, Ambleside, UK, pp. 365–370, July. 2001.
[5] J. Fan, “Array codes as low-density parity-check codes,” in Proc. 2nd Int. Symp. on Turbo Codes & Related Topics, Brest, France, 2000.
[6] M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1788–1793, Aug. 2004.
[7] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory, vol. IT-27, no. 5, pp. 533–547, Sep. 1981.
[8] R. M. Tanner et. al., “LDPC Block and Convolutional Codes Based on Circulant Matrices,” IEEE Trans. Information Theory, vol. 50, NO. 12, pp. 2966-2984, Dec. 2004.
[9] J. L. Kim et. al., “Explicit Construction of Families of LDPC Codes With No 4-Cycles,” IEEE Trans. Information Theory, vol. 50, NO. 10, pp. 2378-2388, Oct. 2004.
[10] L. Chen, J. Xun, I. Djurdjevic, and S. Lin, “Near Shannon limit quasicyclic low density parity-check codes,” IEEE Trans. Commun., vol. 52, no. 7, pp. 1038–1042, Jul. 2004.
[11] Z. Li, L. Chen, L. Zeng, S. Lin, and W. Fong, “Efficient encoding of quasi-cyclic low-density parity check codes,” IEEE Trans. Commun., vol. 54, no. 1, pp. 71–81, Jan. 2006.
[12] T. J. Richardson and R. L. Urbanke, “Efficient encoding of Low-Density Parity-Check codes,” IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 638-656, Feb. 2001.
[13] C. M. Huang, J. F. Huang and C. C. Yang, “Construction of QC-LDPC Codes from Quadratic Congruences,” IEEE Commun. Lett., vol. 12, no. 4, pp. 313-315, Apr. 2008.
[14] Z. Kostic and E. L. Titlebaum, “The design and performance analysis for several new classes of codes for optical synchronous CDMA and for arbitrary-medium time-hopping synchronous CDMA communication systems,” IEEE Trans. Commun., vol. 42, no. 8, pp. 2608-2617, Aug. 1994.
[15] C. C. Yang, “Optical CDMA Passive Optical Network Using Prime Code With Interference Elimination,” IEEE Photon. Technol. Lett., vol. 32, no. 7, pp. 516-518, Apr. 2007.
[16] N. Miladinovic and M. Fossorier, “Systematic recursive construction of LDPC codes,” IEEE Commun. Lett., vol. 8, no. 5, pp. 302-304, May. 2004.
[17] O. Miladinovic, N. Kashyap and D. Leyla, “Shortened Array Codes of Large Girth,” IEEE Trans. Inform. Theory, vol. 52, no. 5, pp. 3707-3722, Aug. 2006.