RS (255, 249) Codec Based on All Primitive Polynomials Over GF(\(2^8\))

  • Jagannath Samanta
  • Jaydeb Bhaumik
  • Soma Barman
  • Sk. G. S. Hossain
  • Mandira Sahu
  • Subrata Dutta
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 470)


Reed–Solomon (RS) codes are generally employed to detect and correct errors in digital transmission and storage systems. The primitive polynomial has a great role to design any RS codes. In this chapter, a RS (255, 249) codec has been designed and implemented based on sixteen primitive polynomials over GF(\(2^8\)) field. The details of theoretical and FPGA synthesis results of the RS (255, 249) codec are presented here. The area in terms of lookup tables and delay of RS (255, 249) codec have been observed for sixteen primitive polynomials. The RS (255, 249) codec based on primitive polynomial, PP3 = \(x^{8} +x^{5} +x^{3} +x^{2} +1\), has consumed lowest area compared to all other primitive polynomials. This codec architecture can be employed in M-ary phase-shift keying modulation scheme and ultra-wideband application.


Galois field Primitive polynomial Reed–Solomon code Phase-shift keying RiBM algorithm and FPGA 


  1. 1.
    S. Lin, D.J. Costello, Error Control Coding: Fundamentals and Applications. Prentice-Hall (1983)Google Scholar
  2. 2.
    S.B. Wicker, V.K. Bhargava, Reed Solomon Codes and Their Applications. IEEE Press (1994)Google Scholar
  3. 3.
    J. Baek, M.H. Sunwoo, Low hardware complexity key equation solver chip for ReedSolomon decoders, in Proceedings of IEEE Asian Solid-State Circuits Conference (Jeju, Korea, 2007), p. 5154Google Scholar
  4. 4.
    I.S. Reed, M.T. Shih, T.K. Truong, VLSI design of inverse-free Berlekamp Massey algorithm, in Proceedings of the Institution of Electrical Engineers, vol. 138 (1991), pp. 295–298Google Scholar
  5. 5.
    H.C. Chang, C.B. Shung, New serial architecture for the Berlekamp Massey algorithm. IEEE Trans. Commun. 47(4), 441–443 (1999)zbMATHGoogle Scholar
  6. 6.
    J.H. Jeng, T.K. Truong, On decoding of both errors and erasures of a Reed-Solomon code using an inverse-free Berlekamp-Massey algorithm. IEEE Trans. Commun. 47(10), 1488–1494 (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    D.V. Sarwate, N.R. Shanbhag, High-speed architectures for Reed-Solomon decoder. IEEE Trans. VLSI Syst. 9(5), 641–655 (2001)CrossRefGoogle Scholar
  8. 8.
    Y.W. Chang, T.K. Truong, J.H. Jeng, VLSI architecture of modified Euclidean algorithm for reed-Solomon code. Inform. Sci. 155(1), 139–150 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Kumar, S. Sawitzki, High Throughput and Low power Reed Solomon Decoder for Ultra Wide Band, in Proceedings of Intelligent Algorithms in Ambient and Biomedical Computing, vol. 7 (2006), pp. 299–316Google Scholar
  10. 10.
    I.S. Jin, Design and implementation of efficient Reed-Solomon decoder for intelligent home networking, in Proceedings of Conference (FGCN), CCIS 265 (2011), pp. 261–268Google Scholar
  11. 11.
    J. Samanta, J. Bhaumik, S. Barman, FPGA based area efficient RS (23, 17) codec. Microsyst. Technol. 23(3), 639–650 (2017)CrossRefGoogle Scholar
  12. 12.
    M.B. Dissanayake, Y. Deng, A. Nallanathan, E.M.N. Ekanayake, M. Elkashlan, Reed Solomon codes for molecular communication with full absorption receiver. IEEE Commun. Lett. (2017)Google Scholar
  13. 13.
    (255, 249) Reed Solomon Decoder, Algorithm data sheet. Blue Rum Consulting Limited (2009)Google Scholar
  14. 14.
    G. Heidari, WiMedia UWB: Technology of Choice for Wireless USB and Bluetooth. Wiley (2008). ISBN: 978-0-470-51834-2Google Scholar
  15. 15.
    R.D. Cideciyan, E. Eleftheriou, Concatenated Reed-Solomon/convolutional coding scheme for data transmission in CDMA cellular systems, in IEEE Vehicular Technology Conference (1994), pp. 1369–1373Google Scholar
  16. 16.
    Kar Peo Yar, Design and analysis of short packet and concatenated coded communication systems, Ph.D. Thesis, The University of Michigan, 2007Google Scholar
  17. 17.
    S.N. Ramlan, R. Mohamad, N. Arbain, Implementation of M-ary Phase Shift Keying (PSK) base band modem on Texas instrument digital signal processor TMS320C6713, in Proceedings in IEEE International Conference on Computer Applications and Industrial Electronics (ICCAIE) (2011), pp. 627–632Google Scholar
  18. 18.
    M. Wan, N. Zhang, Searching IP Blocks: Application Specific Components,
  19. 19.
    D. Bhattacharya, D. Mukhopadhyay, D. Roy, Chowdhury, A cellular automata based approach for generation of large primitive polynomial and its application to RS-Coded MPSK modulation. Lecture Notes in Computer Science vol. 4173 (2006), pp. 204–214Google Scholar
  20. 20.
    T. Hansen, G.L. Mullen, Primitive polynomials over finite fields. Math. Comput. 59(200), 639–643 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    W. Han, The distribution of coefficient of primitive polynomials over finite fields. Cryptogr. Comput. Number Theory 20, 43–57 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    C. Xiaojun, G. Jun, L. Zhihui, RS encoder design based on FPGA, in Proceedings of 2nd IEEE, ICACC 2010, vol. 1 (2010), pp. 419-421Google Scholar
  23. 23.
    R. Huynh, G.E. Ning, Y. Huazhong, A low power error detection in the syndrome calculator block for Reed-Solomon codes: RS(204, 188). J. Tsinghua Sci. Technol. 14(4), 474–477 (2009)CrossRefGoogle Scholar
  24. 24.
    J.I. Park, K. Lee, C.S. Choi, H. Lee, High-speed low-complexity Reed-Solomon decoder using pipelined Berlekamp-Massey algorithm and its folded architecture. J. Semicond. Technol. Sci. 10(3), 193–202 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Jagannath Samanta
    • 1
  • Jaydeb Bhaumik
    • 1
  • Soma Barman
    • 2
  • Sk. G. S. Hossain
    • 3
  • Mandira Sahu
    • 1
  • Subrata Dutta
    • 1
  1. 1.Haldia Institute of TechnologyHaldiaIndia
  2. 2.Institute of Radio Physics and ElectronicsUniversity of CalcuttaKolkataIndia
  3. 3.Department Of CSEAliah UniversityNewtownIndia

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