A fast parallel implementation of the Berlekamp-Massey algorithm with a 1D systolic array architecture

  • Shojiro Sakata
  • Masazumi Kurihara
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


In this paper we present a fast parallel version of the BM algorithm based on a one-dimensional (1D) or linear systolic array architecture which is composed of a series of m cells (processing units), where m is the size of the given data, i.e., the length of the input sequence. The 1D systolic array has only local communication links between each two neighboring cells without any global or nonlocal links between distant cells. Each cell executes a small fixed number of operations at every time unit. Our implementation with the 1D systolic array architecture attains time complexity \(\mathcal{O}\left( m \right)\) so that we can have the optimal total complexity \(\mathcal{O}\left( {m^2 } \right)\), which means that both requirements of (1) maximum throughput rate and of (2) local communication are satisfied, as is the case with some fast parallel implementations of the extended Euclidean algorithm. Our method gives not only another proof of equivalence between the Berlekamp-Massey algorithm and the extended Euclidean algorithm, in particular in the realm of parallel processing, but also alternatives of practical and efficient architectures for R.S. decoders.


Systolic Array Toeplitz System Systolic Architecture Efficient Parallel Algorithm Extended Euclidean Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Sun-Yuan Kung and Yu Hen Hu, “A highly concurrent algorithm and pipelined architecture for solving Toeplitz systems”, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.31, pp.66–76, 1983.Google Scholar
  2. [2]
    J.L. Dornstetter, “ On the equivalence between Berlekamp's and Euclid's algorithm”, IEEE Transctions on Information Theory, vol.33, pp.428–431, 1987.Google Scholar
  3. [3]
    R.P. Brent and H.T. Kung, “Systolic VLSI arrays for polynomial GCD computation”, Technical Report, Carnegie-Mellon University, Computer Science Department, May 1982.Google Scholar
  4. [4]
    Howard M. Shao, T.K. Truong, Leslie J. Deutsch, Joseph H. Yun, and Irving S. Reed, “A VLSI design of a pipelined Reed-Solomon decoder,” IEEE Transactions on Computers, vol.34, pp.393–403, 1985.Google Scholar
  5. [5]
    Keiichi Iwamura, Hideki Imai, and Yasunori Dohi, “A construction method for Reed-Solomon codes suitable for VLSI design” (in Japanese), Transactions of IEICE, vol.J71-A, pp.751–759, 1988.Google Scholar
  6. [6]
    Richard P. Brent and Frnaklin T. Luk, “A systolic array for the linear-time solution of Toeplitz systems of equations”, Journal of VLSI and Computer Systems, vol.1, pp.1–22, 1991.Google Scholar
  7. [7]
    Christopher J. Zarowski, “Schur algorithm for Hermitian Toeplitz and Hankel matrices with singular leading principle submatrices” IEEE Transactions on Signal Processing, vol.39, pp.2464–2480, 1991.Google Scholar
  8. [8]
    Christopher J. Zarowski, “A divisionless form of the Schur Berlekamp-Massey algorithm”, Information Theory and Applications: Proceedings of the Third Canadian Workshop, Ontario (ed. T. Aaron Gulliver and Norman P. Secord), Springer Verlag, pp.38–44, 1994.Google Scholar
  9. [9]
    Keiichi Iwamura and Hideki Imai, “A method to find a minimal set of polynomials capable of generating a given finite N-dimensional array” (in Japanese), Transactions of IEICE, vol.J77-A, pp.982–991, 1994.Google Scholar
  10. [10]
    Shojiro Sakata, Jørn Justesen, Yvonne Madelung, Helge Elbrønd Jensen, and Tom Høholdt, “Fast decoding of AG codes up to half the designed minimum distance”, submitted for IEEE Transactions on Information Theory.Google Scholar
  11. [11]
    Shojiro Sakata, Jørn Justesen, Yvonne Madelung, Helge Elbrønd Jensen, and Tom Høholdt, “A fast decoding method of AG codes from Miura-Kamiya curves up to half the Feng-Rao bound”, Finite Fields and Their Applications, vol.1, pp.83–101, 1995.Google Scholar
  12. [12]
    Masazumi Kurihara and Shojiro Sakata, “A fast parallel decoding algorithm for one-point AG codes with a systolic architecture”, Technical Report of IEICE, vol.IT94-63, pp.49–56, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Shojiro Sakata
    • 1
  • Masazumi Kurihara
    • 1
  1. 1.Department of Computer Science and Information MathematicsThe University of Electro-CommunicationsTokyoJapan

Personalised recommendations