論文摘要
這本論文中我們提出建構周長(girth)任意大的低密度同位元檢查碼(low density parity check code)的方法，並探討位元翻轉演算法(bit flipping algorithm)更正錯誤的能力與碼的周長的關係。

論文的第一部份，我們以遞迴的方式應用克羅內克積(Kronecker product)建構低密度同位元檢查碼。這種方法所建構的錯誤更正碼，其同位元檢查矩陣由循環排列矩陣(circulant permutation matrix)所組成，是半循環碼(quasi-cyclic code)的一種，具有可快速編碼的特性。我們研究文獻中的LU(m,q)碼，發現其建構的方式，可以用克羅內克積表示，這表示LU(m,q)碼也具有可快速編碼的特性。但對於某些參數m, q, LU(m,q)碼所對應的Tanner圖是不相連的，會造成碼的效能下降，因此我們也推導出Tanner圖是相連的條件。

建構已知周長的低密度同位元檢查碼之後，我們分析一種貪婪式位元翻轉解碼器的錯誤更正能力。位元翻轉解碼器具有較低的複雜度，且結構簡單，適用於高流通量(throughput)的應用。藉由定義一個變數節點相鄰圖(variable node adjacency graph)，我們可以估算每個變數節點的非零的檢查和(check sum)，而此變數節點相鄰圖是由Tanner圖推導出來的。我們推導出的錯誤更正能力與周長g的關係為：當碼的周長為8時，貪婪式位元翻轉解碼器可以解lambda-1個錯誤，其中的lambda是碼的行權重(column weight)，在相同的條件下，蓋勒格(Gallager)原始的位元翻轉解碼器只能保證解lambda/2個錯誤。這個結果顯示對某一些碼，貪婪式位元翻轉解碼器在二元對稱通道(binary symmetric channel)可以達到隨機錯誤更正能力(random error correcting capability)。至於對周長大於等於10的情形，我們僅證明其解碼能力為(1/2)*n_0(lambda/2, g/2)，和蓋勒格原始的位元翻轉解碼器相同，此結果尚有改善空間, 其中n_0(,)是圖論中的莫耳下界(Moore bound)。

In the first part of this dissertation, we apply generalized Kronecker product recursively
to construct low-density parity-check (LDPC) codes with arbitrarily
large girth. The parity-check matrices of these codes
are block matrices consisting of circulant permutation matrices,
thus may benefit from efficient encoding.
It turns out that the LU(m, q) codes proposed in the literature
is a special case of this construction for prime q.
Connectivity of Tanner graphs of these codes are investigated.
The performance of codes derived from LU(m, q) codes are demonstrated.

The second part, the error correction capability of
a greedy bit-flipping (BF) decoding algorithm
for LDPC codes
is studied by introducing variable node adjacency
graphs which are derived from Tanner graphs
of LDPC codes.
For codes with column weight lambda and girth g=8,
it can be shown that error patterns
of weight less than or equal to lambda-1 can be corrected,
while Gallager's BF algorithm can correct lambda/2 errors.
This result implies that the greedy BF algorithm can
decode up to the random error-correcting capability over binary symmetric channels
for girth 8 codes with minimum distance 2*lambda.

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