LDPC(Low-Density Parity-Check)碼為下個世代的先進通訊標準所採用的錯誤更正碼，其優異的錯誤更正能力可以逼近Shannon的理論值，配合訊息傳遞(message passing, MP)演算法，可以快速得到傳送端所發出之訊息。大部分好的LDPC碼是利用電腦產生的，特別是碼長較長的LDPC碼。但由於不具有特殊的結構，故在編碼端的複雜度偏高。Kou, Lin, Fossorier [13]，基於有限幾何，首先提出系統化的方法建構LDPC碼。如此所建構之眾多類的LDPC碼具有良好的最小漢明距離的特性；且Tanner圖裡並不包含短迴圈。另外，在編碼部分也較為簡單，且可由線性暫存器在實現。基於上述所使用的建構方法，對於有限幾何附加其它的限制條件來建構其它的LDPC碼。考慮三維投影空間裡的非退化二次式，, 利用二次式所擁有的特殊性質，針對某些參數，嘗試利用數學的方式去證明。 LDPC code used by the advanced communication standard of the next generation is an error control code. Its error correction ability may approach the Shannon’s theoretical value. With the MP algorithm, it can decode received samples in high speed from transmitter. Good LDPC codes that have been found are largely computer generated, especially long codes, and their encoding is very complex owing to the lack of structure. Kou, Lin, Fossorier [13] introduced the first algebraic and systematic construction of LDPC codes based on finite geometries. The large classes of finite-geometry LDPC codes have relatively good minimum distances, and their Tanner graphs do not contain short cycles. Consequently, their encoding is simple and can be implemented with linear shift registers. Based on the above construction method on finite geometries[13], we append more restrictions on finite geometries to construct LDPC codes using the non-degenerated quadratic surfaces on three-dimensional projective geometry. Owing some special properties on quadratic surfaces, some parameters of LDPC can be proven mathematically.
