This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
A Systolic Array Implementation of the Feng-Rao Algorithm
July 1999 (vol. 48 no. 7)
pp. 690-706

Abstract—An efficient implementation of a parallel version of the Feng-Rao algorithm on a one-dimensional systolic array is presented in this paper by adopting an extended syndrome matrix. Syndromes of the same order, lying on a slant diagonal in the extended syndrome matrix, are scheduled to be examined by a series of cells simultaneously and, therefore, a high degree of concurrency of the Feng-Rao algorithm can be achieved. The time complexity of the proposed architecture is $m+g+1$ by using a series of $t+\lfloor {\frac{g-1}{2}} \rfloor +1$, nonhomogeneous but regular, effective processors, called PE cells, and $g$ trivial processors, called D cells, where $t$ is designed as the half of the Feng-Rao bound. Each D cell contains only delay units, while each PE cell contains one finite-field inverter and, except the first one, one or more finite-field multipliers. Cell functions of each PE cell are basically the same and the overall control circuit of the proposed array is quite simple. The proposed architecture requires, in total, $t+\lfloor {\frac{g-1}{2}} \rfloor +1$ finite-field inverters and ${\frac{(t+\lfloor (g-1)/2 \rfloor)(t+\lfloor (g-1)/2 \rfloor +1)}{2}}$ finite-field multipliers. For a practical design, this hardware complexity is acceptable.

[1] V.D. Goppa, “Codes Associated with Divisors,” Probl. Peredachi Inform., vol. 13, no. 1, pp. 33-39, 1977.
[2] V.D. Goppa, “Codes on Algebraic Curves,” Dokl. Akad. Nauk SSSR, vol. 24, pp. 170-172, 1981.
[3] V.D. Goppa, “Algebraic-Geometric Codes,” Izv. Akad. Nauk. SSSR, vol. 21, pp. 75-91, 1983.
[4] M.A. Tsfasman, S.G. Vladut, and T. Zink, “Modular Curves, Shimura Curves, and Goppa Codes Better Than Varshamove-Gilbert Bound,” Math. Nachr., vol. 104, pp. 13-28, 1982.
[5] G.D. Forney Jr., “On Decoding BCH Codes,” IEEE Trans. Information Theory, vol. 11, pp. 549-557, Oct. 1965.
[6] E.R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
[7] J. Massey,“Shift-register synthesis and BCH decoding,” IEEE Trans. on Information Theory, vol. 15, pp. 122-127, 1969.
[8] S. Sakata, “Highly Concurrent Parallel Version of the Berlekamp-Massey Algorithm and Its Pipelined Architecture,” Proc. ISITA '94, pp. 1,069-1,072, Sydney, Australia, 1994.
[9] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. New York: North-Holland, 1978.
[10] H. Stichtenoth, Algebraic Function Fields and Codes. New York: Springer-Verlag, 1993.
[11] M.A. Tsfasman and S.G. Vladut, “Geometric Approach to Higher Weights,” IEEE Trans. Information Theory, vol. 41, pp. 15,64-1,588, Nov. 1995.
[12] T. Høholdt and R. Pellikaan, “On the Decoding of Algebraic-Geometric Codes,” IEEE Trans. Information Theory, vol. 41, pp. 1,589-1,614, Nov. 1995.
[13] J. Justesen, K.J. Larsen, H.E. Jensen, A. Havemose, and T. Høholdt, “Construction and Decoding of a Class of Algebraic Geometry Codes,” IEEE Trans. Information Theory, vol. 35, pp. 811-821, May 1989.
[14] G.-L. Feng and T.R.N. Rao, “Decoding Algebraic-Geometric Codes Up to the Designed Minimum Distance,” IEEE Trans. Information Theory, vol. 39, pp. 37-45, Jan. 1993.
[15] G. Strang, Linear Algebra and Its Applications, third ed. San Diego, Calif.: Harcourt Brace Jova novich, 1988.
[16] W. Fulton, Algebraic Curves. New York: W.A. Benjamin, 1969.
[17] G.-L. Feng, V.K. Wei, T.R.N. Rao, and K.K. Tzeng, “Simplified Understanding and Effect Decoding of a Class of Algebraic-Geometric Codes,” IEEE Trans. Information Theory, vol. 40, pp. 981-1,002, July 1994.
[18] K. Saints and C. Heegard, “Algebraic-Geometric Codes and Multidimensional Cyclic Codes: A Unified Theory and Algorithms for Decoding Using Gröbner Bases,” IEEE Trans. Information Theory, vol. 41, pp. 1,733-1,751, Nov. 1995.
[19] S. Sakata, J. Justesen, Y. Madelung, H.E. Jensen, and T. Hohøldt, “A Fast Decoding Method of AG Codes from Miura-Kamiya Curves$C_{ab}$Up to Half the Feng-Rao Bound,” Finite Fields and Their Applications, vol. 1, pp. 83-101, 1995.
[20] M. Kurihara and S. Sakata, “A Fast Parallel Decoding Algorithm for General One-Point AG Codes with a Systolic Array Architecture,” Proc. IEEE ISIT '95, p. 99, 1995.
[21] S. Sakata and M. Kurihara, “A Systolic Array Architecture for Implementation a Fast Parallel Decoding Algorithm of One-Point AG Codes,” Proc. IEEE ISIT '97, p. 378, 1997.
[22] M.E. O'Sullivan, “VLSI Architecture for a Decoder for Hermitian Codes,” Proc. IEEE ISIT '97, p. 376, 1997.
[23] R. Kötter, “Fast Generalized Minimum-Distance Decoding of Algebraic-Geometry and Reed-Solomon Codes,” IEEE Trans. Information Theory, vol. 42, pp. 721-737, May 1996.
[24] A. Skorobogatov and S. Vladut, “On the Decoding of Algebraic-Geometric Codes,” IEEE Trans. Information Theory, vol. 36, pp. 1,051-1,060, Sept. 1990.
[25] H.-T. Kung, “Why Systolic Architectures?,” Computer, vol. 15, no. 1, pp. 37-46, Jan. 1982.
[26] S.Y. Kung, VLSI Array Processors. Prentice Hall, 1988.
[27] P.-S. Tseng, A Systolic Array Parallelizing Compiler. Boston: Kluwer Academic, 1990.
[28] B. Hochet,P. Quinton, and Y. Robert,"Systolic Gaussian Elimination overGF(p) with Partial Pivoting," IEEE Trans. Computers, vol. 38, no. 9, pp. 1,321-1,324, Sept. 1989.
[29] G.-L. Feng and T.R.N. Rao, “A Simple Approach for Construction of Algebraic-Geometric Codes from Affine Plane Curves,” IEEE Trans. Information Theory, vol. 40, pp. 1,003-1,012, July 1994.

Index Terms:
Error-correcting codes, algebraic-geometric codes, Feng-Rao algorithm, systolic array.
Citation:
Chih-Wei Liu, Kuo-Tai Huang, Chung-Chin Lu, "A Systolic Array Implementation of the Feng-Rao Algorithm," IEEE Transactions on Computers, vol. 48, no. 7, pp. 690-706, July 1999, doi:10.1109/12.780877
Usage of this product signifies your acceptance of the Terms of Use.