張耀祖教授及李崇道教授在2010年，提出了使用拉格朗日插值公式的代數解碼法來解二元循環碼，但並未提出一個有效的方法來計算拉格朗日插值公式。本篇論文是使用最低公倍式的概念來計算在解碼過程中需要的拉格朗日插值公式。
令E是體F = F_p上維度為m的擴充體、β是E中乘法單位元素的n次本元根、m_r(x)代表E中元素r在F中的最小多項式。則我們將廣義最小多項式Min (r, F) 定義為m_rβ^i(x)，i從0到n-1的最低公倍式。本篇論文討論的二元平方剩餘碼中的所有可解的錯會是某些廣義最小多項式的根，所有可解錯的集合將會被這些廣義最小多項式的根的集合所分割。基於這個想法，我們發展一個有效率的方法來計算拉格朗日插值公式。

In 2010, Prof. Chang and Prof. Lee applied Lagrange interpolation formula to decode a class of binary cyclic codes, but they did not provide an effective way to calculate the Lagrange interpolation formula. In this thesis, we use the least common multiple of polynomials to compute it effectively.
Let E be an extension field of degree m over F = F_p and β be a primitive nth root of unity in E. For a nonzero element r in E, the minimal polynomial of r over F is denoted by m_r(x). Then, let Min (r, F) denote the least common multiple of m_rβ^i(x) for i = 0, 1,..., n-1 and be called the generalized minimal polynomial of over F. For any binary quadratic residue code mentioned in this thesis, the set of all its correctable error patterns can be partitioned into root sets of some generalized minimal polynomials over F. Based on this idea, we can develop an effective method to calculate the Lagrange interpolation formula.

論文審定書 i
誌謝 ii
摘要 iii
Abstract iv
1 Introduction 1
2 Finite Fields 3
2.1 Existence of Finite Fields 3
2.2 Basic Properties 6
2.2.1 Cyclic group 6
2.2.2 Subfield 6
2.2.3 Minimal polynomial 7
3 Generalized Minimal Polynomial 9
4 Application in Coding Theory 14
4.1 Quadratic Residue Code 14
4.2 Lagrange Interpolation Formula 17
4.3 Algebraic Decoding Algorithm 21
5 Conclusion 25
Bibliography 26

[1] G. L. Feng and K. K. Tzeng, “A new procedure for decoding cyclic and BCH codes up to actual minimum distance,” IEEE Trans. Inform. Theory, vol. 40, no. 5. pp. 1364-1374, Sept. 1994.
[2] I. S. Reed, M. T. Shih, and T. K. Troung, “VLSI design of inverse-free Berlekamp-Massey algorithm,” IEE PROCEEDINGS-E, vol. 138, No. 5, pp. 295-298, Sept. 1991.
[3] R. T. Chien, “Cyclic decoding procedures for the Bose-Chaudhuri-Hocquenghem codes", IEEE Trans. Inform. Theory, vol. 10, no. 4, pp. 357–363, Oct. 1964.
[4] Y. Chang and C. D. Lee, “Algebraic decoding of a class of binary cyclic codes via Lagrange interpolation formula.” IEEE Trans. Inform. Theory, vol. 56, no. 1, pp. 130-139, Jan. 2010.
[5] 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, Sept. 2003.
[6] E. R. Berlekamp, “Algorithmic Coding Theory,” McGraw-Hill, New York, 1968.
[7] R. Lidl and H. Niederreiter, Finite Fields. Addison-Wesley, Reading, MA, 1983; Cambridge Univ. Press, Cambridge, UK, 11, 1997.