近年來,低密度同位元檢查碼(Low Density Parity Check Code)搭配和積演算法(Sum-Product Algorithm)展現了接近薛氏極限(Shannon Limit)的效能,因而受到廣泛的重視.低密度同位元檢查碼被視為渦輪碼(Turbo Code)的最主要對手.然而和積演算法是一個高硬體複雜度的演算法,這項限制使得和積演算法難以實現.在這篇論文裡,我們提出一個低硬體複雜度的解碼演算法來實現低密度同位元檢查碼的解碼.在低密度同位元碼中,其同位元檢查矩陣的非零元素相當稀疏,數目約正比於碼字(code-word)的長度,因此在低密度同位元碼的碼字中,每個位元的相關聯位元較一般的區塊碼(block code)少了許多.我們在此演算法中利用這個特性,使用了一個低複雜度的方法來決定一個抹失位元(erasure bit)的原始值,並讓此被決定的抹失位元提共更多的資訊以決定其他的抹失位元.以此決定抹失的機制為基礎,配合一個低複雜度的程序來處理剩餘的錯誤位元(error bit),我們提出一個低複雜度的解碼演算法.此演算法只使用相當簡單的運算單元,並且不像和積演算法需要龐大的記憶體來儲存演算過程中傳遞的訊息(message passing),因此是一個低硬體複雜度的演算法,較易於實現.我們提出的演算法屬於循序式的(sequential)的,其演算複雜度正比於碼字長度(code-word length).相較於一般區塊碼,其複雜度與碼字長度成非線性比例.此優勢使得我們的演算法可適用於非常長的低密度同位元碼.

In recent years, low-density parity-check (LDPC) codes catch high attentions because of
its near Shannon limit performance with sum-product decoding algorithm. LDPC codes are
regarded as the main competitor of turbo codes. However, sum-product algorithm is a high
hardware complexity algorithm. This restriction makes LDPC codes hard to realize. In this
thesis, we propose a low hardware complexity algorithm to realize the decoding of LDPC
code by modelling the AWGN channel as a noisy erasure channel. In the proposed decoding
algorithm, we basically use all-but-one rule to decode erasures while carefully managing the
possibly noisy hard decisions. Since this algorithm takes only very simple operation units
and does not need a huge number of memories, it can be applies to decode very long codes,
which may be attractive in practical situations.

[1] R. Gallager, Low-density Parity-check Codes. Cambridge,MA: MIT Press, 1963.
[2] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, \Factor graphs and the sum-product
algorithm," IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 498{519, Feb. 2001.
[3] S. M. Aji and R. J. McEliece, \The generalized distributive law," IEEE Trans. Inform.
Theory, vol. 46, no. 2, pp. 325{343, Mar. 2000.
[4] M. Davey and D. MacKay, \Low-density parity-check codes over GF(q)," IEEE Com-
munications Letters, vol. 2, no. 6, pp. 165{167, Jun. 1998.
[5] R. Tanner, \A recursive approach to low complexity codes," IEEE Trans. Inform. The-
ory, vol. IT-27, pp. 533{547, Sept. 1981.
[6] D. MacKay, \Good error-correcting codes based on very sparse matrices," IEEE Trans.
Inform. Theory, vol. 45, pp. 399{431, Mar. 1979.
[7] Y. Kou, S. Lin, and M. P. Fossorier, \Low-density parity-check codes based on nite
geometries: a rediscovery and new results," IEEE Trans. Inform. Theory, vol. 47, no. 7,
pp. 2711{2736, Nov. 2001.
[8] T. Richardson, M. Shokrollahi, and R. Urbanke, \Design of capacity-approaching irreg-
ular low-density parity-check codes," IEEE Trans. Inform. Theory, vol. 47, no. 2, pp.
619{637, Feb. 2001.
[9] M. Luby, M. Mitzenmacher, M. Shokrollahi, and D. Spielman, \Improved low-density
parity-check codes using irregular graphs," IEEE Trans. Inform. Theory, vol. 47, no. 2,
pp. 585{598, Feb. 2001.
[10] M. Sipser and D. Spielman, \Expander codes," IEEE Trans. Inform. Theory, vol. 42,
no. 6, pp. 1710{1722, Nov. 1996.
[11] D. MacKay, S. Wilson, and M. Davey, \Comparison of constructions of irregular gallager
codes," IEEE Trans. Comm., vol. 47, no. 10, pp. 1449{1454, Oct. 1998.
[12] T. Richardson and R. Urbanke, \Ecient encoding of low-density parity-check codes,"
IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 638{656, Feb. 2001.
[13] M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, \Ecient erasure
correcting codes," IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 569{584, Feb. 2001.