在本論文中，我們提出了一個在實作上可達到低硬體複雜度的赫米碼的編解碼器硬體架構。藉由利用模數（module）的Gröbner 基底的理論，我們發展了一個和傳統的循環碼編碼器類似的序列輸入及序列輸出的有系統的（systematic）赫米碼編碼器。藉著推導出記憶體和常數乘法器數目的上限，我們可證明我們的編碼器架構的複雜度，遠比直接用矩陣乘法的方法所建立的編碼器的複雜度低很多。
如同BCH碼或理德所羅門碼（Reed-Solomon code）的解碼，我們可將赫米碼的解碼程序分成以下四個步驟：從接收到的訊號產生已知的特徵值（known syndrome）、產生錯誤位置多項式（error-locator polynomial）、搜尋錯誤位置（error location）以及計算錯誤值（error value）。
藉由在有限場GF(q2)上赫米曲線Hq (x,y) 上所有的有限有理點的 (x, y) 座標值的規則代數性質，我們可以擴展在理德所羅門碼的解碼中所用到的Horner’s 規則以及Chien 搜尋的架構，提出可用於赫米碼的特徵值產生器以及錯誤位置搜尋器的有效率的硬體架構。
在產生錯誤位置多項式方面，我們採用Liu-Lu演算法，經由一維的心臟收縮式陣列（systolic array）的結構，來發展出一個可搜尋出在無窮遠處的有理點Q∞ 上有最小極點序（pole order）的錯誤位置多項式的硬體架構。假設γ為赫米曲線裡最小的非零非缺口數（nongap），此處γ等於有理函數x 在在無窮遠處的有理點Q∞ 上的極點序ox。藉由利用延伸的赫米碼特徵矩陣（extendedsyndrome matrix）以及γ-轉移性質（γ-shift properties），在Liu-Lu演算法裡，只有延伸的赫米碼特徵矩陣上的γ行必須被同時檢查到。因此在我們的架構中，我們只需要 個稱為PE細胞（PE cell）的程序處理元素（processing element）、g個稱為D細胞（D cell）的延遲單元（delay unit）以及γ個稱為OE細胞（OE cell）的運算元素（operation element），此處g為赫米曲線的屬（genus）、 為赫米碼 中被設計的的容錯能力（designed error correcting capability）。
最後，在解錯誤值方面，我們更改了Hansen演算法，提出了一個只有一個錯誤位置多項式已知的條件下，仍能解出錯誤值的演算法。並依據我們所提出的演算法，藉由一維的心臟收縮式陣列結構，發展出一個可解出錯誤值的有效率的硬體架構。

In this thesis,
a codec architecture of Hermitian codes
which is able to target a low hardware complexity of implementation
is presented.
By exploiting the theory of Grobner bases for modules,
we develop a serial-in-serial-out hardware architecture,
similar to a classical cyclic encoder,
for the systematic encoding scheme of Hermitian codes.
By deriving the upper bounds of the numbers of memory elements
and constant multipliers in the proposed architecture,
we show that the complexity of our architecture
is much less than that of the brute-force systematic encoding
by matrix multiplication.
Similar to the decoding of Reed-Solomon codes or BCH codes,
we divide the procedure for the decoding of Hermitian codes
into four steps:
generating known syndromes from a received word,
producing an error-locator polynomial,
searching for the potential error points (positions)
and evaluating the error values.
Based on the regular algebraic properties
of the (x,y)-coordinates of all finite rational points
on the Hermitian curve ,
we extend the use of Horner's rule and the mechanism of Chien search
in the decoding of Reed-Solomon codes
to render up efficient architectures
for syndrome generation and error location search.
By adopting Liu-Lu algorithm,
we present a hardware architecture
for finding the error-locator polynomial with least pole order at Q,
where Q is the rational point at infinity of the Hermitian curve,
via one-dimensional systolic arrays.
With the extended syndrome matrix M and the r-shift properties,
where r is the smallest nonzero nongap of the Hermitian curve,
only r columns will be examined simultaneously in Liu-Lu algorithm.
Thus only a series of t+(g-1)/2+1
processing elements, called PE cells,
g delay units, called D cells,
and r operation elements, called OE cells,
are needed in our architecture,
where g is the genus of the Hermitian curve and
t is the designed error correcting capability
of the Hermitian code .
Finally, we modify the Hansen's algorithm to fit our case
that only one error-locator polynomial is given
and propose an efficient architecture via one-dimensional systolic arrays
for the determination of the error values.

1] V. D. Goppa, \Codes associated with divisors," Probl. Peredachi Inform., vol. 13,
no. 1, pp. 33{39, 1977.
[2] V. D. Goppa, \Algebraic-geometric codes," Izv. Akad. Nauk SSSR, vol. 21, pp. 75{91,
1983.
[3] J. L. C. Heegard and K. Saints, \Systematic encoding via grobner bases for a class of
algebraic-geometric Goppa codes," IEEE Trans. Inform. Theory, vol. 41,
pp. 1752{1761, Nov. 1995.
[4] J. Little, K. Saints, and C. Heegard, \On the structure of Hermitian codes," Pure and
Applied Algebra, pp. 293{314.
[5] B.-Z. Shen, \On encoding and decoding of the codes from Hermitian curves,"
Cryptography and Coding III, IMA Conf. Proc. Ser., pp. 337{356, 1993.
[6] H. Stichtenoth, \A note on Hermitian codes over gf(q2)," IEEE Trans. Inform.
Theory, vol. 34, pp. 1345{1348, Sep. 1988.
[7] M. E. O'Sullivan, \VLSI architecture for a decoder for Hermitian codes," Proceedings
of ISIT'97, p. 376.
[8] H. J. Tiersma, \Remarks on codes from Hermitian curves," IEEE Trans. Inform.
Theory, vol. 33, pp. 605{607, Jul. 1987.
[9] J. H. van Lint and T. A. Springer, \Generalized Reed-Solomon codes from algebraic
geometry," IEEE Trans. Inform. Theory, vol. 33, pp. 305{309, 1987.
[10] T. Yaghoobian and I. F. Blake, \Hermitian codes as generalized Reed-Solomon codes,"
Designs, Codes and Cryptography, vol. 2, pp. 5{17, 1992.
[11] C.-W. Liu, A Simple and Ecient Decoding Algorithm for Algebraic-Geometric Codes.
PhD thesis, National Tsing Hua University, Taiwan, Jul. 1999.
[12] G.-L. Feng and T. R. N. Rao, \Decoding algebraic-geometric codes up to the designed
minimum distance," IEEE Trans. Inform. Theory, vol. 39, pp. 37{45, Jan. 1993.
[13] G.-L. Feng, V.-K. Wei, T. R. N. Rao, and K.-K. Tzeng, \Simpli ed understanding and
e ect decoding of a class of algebraic-geometric codes," IEEE Trans. Inform. Theory,
vol. 40, pp. 981{1002, Jul. 1994.
[14] M. Kurihara and S. Sakata, \A fast parallel decoding algorithm for general one-point
AG codes with a systolic array architecture," Proceedings of IEEEE ISIT'95, p. 99.
[15] R. Kotter, \A fast parallel implementation of a Berlekamp-Massey algorithm for
algebraic-geometric codes," IEEE Trans. Inform. Theory, vol. 44, pp. 1353{1368, Jul.
1998.
[16] C.-W. Liu, K.-T. Huang, and C.-C. Lu, \A systolic array implementation of the
Fang-Rao algorithm," IEEE Trans. Computers, vol. 48, pp. 690{706, Jul. 1999.
[17] S. Sakata and M. Kurihara, \A systolic array architecture for implementation a fast
parallel decoding algoritm of one-point AG codes," Proceedings of ISIT'97, p. 378.
[18] C.-W. Liu and C.-C. Lu, \An excursion from gaussian elimination to early stoped
Berlekamp-Massey algorithm." submitted for publication.
[19] R.-T. Chien, \Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem
codes," IEEE Trans. Inform. Theory, vol. 10, pp. 357{363, Oct. 1964.
[20] S. Lin and J. D. J. Costello, Error Control Coding: Fundamentals and Applications.
New Jersey: Prentice Hall, 1983.
[21] J. Justensen, K. J. Larsen, H. E. Jensen, and T. Hholdt, \Fast decoding of codes from
algebraic plane curves," IEEE Trans. Inform. Theory, vol. 38, pp. 111{119, Jan. 1992.
[22] D. A. Leonard, \A generalized Forney formula for algebraic-geometric codes," IEEE
Trans. Inform. Theory, vol. 42, pp. 1263{1268, Jul. 1996.
[23] D. A. Leonard, \Ecient Forney fuctions for decoding AG codes," IEEE Trans.
Inform. Theory, vol. 45, pp. 260{265, Jan. 1999.
[24] H. E. J. J. P. Hansen and R. Kotter, \Determination of error values for
algebraic-geometry codes and the forney formula," IEEE Trans. Inform. Theory,
vol. 44, pp. 1881{1886, Sep. 1998.
[25] W. Fulton, Algebraic Curves. New York: W. A. Benjamin Inc., 1969.
[26] H. Stichtenoth, Algebraic Function Fields and Codes. New York: Springer-Verlag,
1993.
[27] W. W. Adams and P. Loustaunau, An Introduction to Grobner Bases. Providence, RI:
Amer. Math. Soc., 1994.
[28] R. Lidl and H. Niederreiter, Finite Fields. Cambridge, UK: Cambridge University
Press, 1997.
[29] H.-S. Wang, \VLSI implementation of syndrome calculator and error locator nder in
the decoding of Hermitian codes," Master's thesis, Department of Electrical
Engineering, National Tsing-Hwa University, R.O.C., Taiwan, 1998.
[30] G.-L. Feng and K.-K. Tzeng, \A generalization of the Berlekamp-Massey algorithm for
multisequence shift-register synthesis with applications to decoding cyclic codes,"
IEEE Trans. Inform. Theory, vol. 37, pp. 1274{1287, Sep. 1991.[31] P. Q. B. Hochet and Y. Robert, \Systolic Gaussian elimination over gf(p) with partial
pivoting," IEEE Trans. Computers, vol. 38, pp. 1321{1324, Sep. 1989.
[32] S.-Y. Kung, VLSI Array Processors. N.J.: Prentice Hall, 1988.
[33] P.-S. Tseng, A Systolic Array Parallelizing Compiler. Boston: Kluwer Academic, 1990.