# 臺灣博碩士論文加值系統

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 近幾年，使用拉格朗日內插法來求一類二元循環碼的未知症狀值的方法已被提出。本論文中，我們提出兩種新的計算方法都是基於症狀值矩陣關係式的尋找及中國餘式定理，使用這些方法來找出未知症狀值的多項式，此多項式變數是用已知症狀值表示。在症狀值矩陣關係式搜尋的部份，我們撰寫程式，對二元循環碼碼長51以下，進行搜尋並列表。針對於二元循環碼，計算表示未知症狀值多項式的部分，將拉格朗日內插法及提出的兩種新方法進行時間的比較。最後，給予一個 (31, 16, 7) 二元平方剩餘碼使用無反根Berlekamp-Massey 的完整解碼例子。
 Recently, the unified unknown syndrome representations to decode a class of binary cyclic codes have been developed by using Lagrange interpolation formula. In this thesis, a new method by combining the syndrome matrix search and Chinese remainder theorem is proposed to express the unified unknown syndrome representation as a rational function in terms of the known syndromes. A computer search has been made to determine the syndrome matrices for binary cyclic codes of lengths less than or equal to 51. Compared to the Lagrange interpolation method, the method presented here substantially reduces the computational time for binary cyclic codes generated by irreducible polynomials. Finally, a complete decoding of the (31, 16, 7) quadratic residue code with inverse-free BerlekampMassey algorithm is given as illustration.
 ACKNOWLEDGMENTS I摘要 IIABSTRACT IIICONTENTS 1LIST OF FIGURES IIILIST OF TABLES IVCHAPTER 1 INTRODUCTION 11.1 HISTORY 11.2 MOTIVATION 21.3 FRAMEWORK 3CHAPTER 2 MATHEMATICAL BACKGROUND 42.1 FINITE FIELD 42.1.1 Definition of a Finite Field 42.1.2 Definition of Field Operations 42.1.3 Prime Finite Field 42.1.4 Finite Field GF(2m) 62.2 ALGEBRAIC CODING THEORY 82.2.1 Cyclic Codes 82.2.2 Quadratic Residue Codes 82.2.3 Encoding of Cyclic Code 92.2.4 Decoding of Cyclic Code 11CHAPTER 3 DECODING METHOD BY LAGRANGE INTERPOLATION 143.1 LAGARANGE INTERPOLATION FORMULA 143.2 UNIFIED REPRESENTATIONS OF UNKNOWN SYNDROME 143.3 INVERSE-FREE BERLEKAMP-MASSEY ALOGRITHM 163.4 CHIEN SEARCH 173.5 DECODING ALGORITHM 17CHAPTER 4 PROPOSED DECODING METHOD 204.1 CHINESE REMAINDER THEOREM 204.2 POLYNOMIAL IN TERMS OF SYNDROMES 214.3 SYNDROME MATRIX 224.4 MAIN RESULT 284.4.1 Modified Chinese Remainder Theorem 284.4.2 Proposed Method 304.4.3 Improved Method 334.5 VARIATIONS 394.6 COMPLEXITY ANALYSIS 43CHAPTER 5 CONCLUSION 44BIBLIOGRAPHY 45APPRNDIX A 47List of FiguresFigure 1 1 The decoding model by general error locator polynomials (left) and unified unknown syndrome representations (right). 3Figure 2 1 Systematic encoding of an (n, k) cyclic code 9Figure 2 2 Simplified model for communication system 11List of TablesTable 2 1 MATHEMATICA OPERICATION OF GF(5) 5Table 2 2 MATHEMATICA OPERICATION OF GF(2) 6Table 2 3 REPRESENTATION OF 7Table 2 4 PRIMITIVE POLYNOMIALS 7Table 2 5 GENERATOR POLYNOMIAL OF BINARY QR CODES. 9Table 3 1 NUMBERS OF (S_1, S_r) FOR QR CODE WITH LENGTH 15Table 4 1 SETS OF SYNDROME MATRICES FOR BINARY 32Table 4 2 ELEMENTS OF 38Table 4 3 COMPUTATIONAL TIME OF PREDETERMINING UNKNOWN SYNDROME REPRESENTATIONS FOR THREE QUADRATIC RESIDUE CODES 43Table A 1 〈K_3〉FOR THE (31, 16, 7) QR CODE 47Table A 2 〈T_3〉FOR THE (31, 16, 7) QR CODE 50
 [1]C. D. Lee, Y. Chang, M. H. Jing, and J. H. Miao, “A new method of predetermining unified unknown syndrome representation for decoding binary cyclic codes,” submit to IET Commun., Apr. 2012 (revised Jul. 2012).[2]E. Prange, “Cyclic Error-Correcting Codes in Two Symbols,” AFCRC-TN-57-103, Air Force Cambridge Research Center, Bedford, Mass., Sept. 1957.[3]E. R. Berlekamp, Algebraic Decoding Theory, New York: McGraw-Hill, 1968.[4]G.-L. Feng and K.-K. Tzeng, “A new procedure for decoding cyclic and BCH codes up to actual minimum distance,” IEEE Trans. Inf. Theory, vol. 40, no. 5, pp. 1364-1374, Sep. 1994.[5]I. S. Reed, M. T. Shih and T. K. Truong, “VLSI design of inverse-free BerleKamp-Massey algorithm,” IEE Proceeding-E, vol. 138, no. 5, pp. 295-298, Sep. 1991[6]J. L. Massey, “Shift-register synthesis and BCH decoding,” IEEETrans. Inf. Theory, vol. IT-15, no. 1, pp. 122-127, Jan. 1969.[7]R. He, I. S. Reed, T. K. Truong, and X. Chen, “Decoding the (47, 24, 11) quadratic residue code,” IEEE Trans. Inf. Theory, vol. 47, no. 3, pp. 1181–1186, Mar. 2001.[8]R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications. New York: Cambridge Univ. Press, 1986.[9]T. K. Truong, P. Y. Shih, W. K. Su, C. D. Lee, and Y. Chang, “Algebraic decoding of (89, 45, 17) quadratic residue code,” IEEE Trans. Inf. Theory, vol. 54, no. 11, pp. 5005-5011, Nov. 2008.[10]T. K. Truong, Y. Chang, Y. H. Chen, and C. D. Lee, “Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes,” IEEE Trans. Commun., vol. 53, no. 5, pp. 749-754, May 2005.[11]Y. Chang and C. D. Lee, “Algebraic decoding of a class of binary cyclic codes via Lagrange interpolation formula,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 130-139, Jan. 2010.[12]Y. Chang, T. K. Truong, I. S. Reed, H. Y. Cheng, and C. D. Lee, “Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes,” IEEE Trans. Commun., vol. 51, no. 9, pp. 1463-1473, Sep. 2003.[13]Y. H. Chen, T. K. Truong, Y. Chang, C. D. Lee, and S. H. Chen, “Algebraic decoding of quadratic residue codes using inverse-free Berlekamp-Massey algorithm,” J. Inf. Sci. Eng., vol. 23, pp. 127-145, Jan. 2007.
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