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Quasi-Cyclic Codes

  • Marco BaldiEmail author
Chapter
Part of the SpringerBriefs in Electrical and Computer Engineering book series (BRIEFSELECTRIC)

Abstract

In this chapter, we recall the main definitions concerning quasi-cyclic codes, which will be used in the remainder of the book. We introduce the class of circulant matrices, and the special class of circulant permutation matrices, together with their isomorphism with polynomials over finite fields. We characterize the generator and parity-check matrices of quasi-cyclic codes, by defining their “blocks circulant” and “circulants block” forms, and show how they translate into an encoding circuit. We define a special class of quasi-cyclic codes having the parity-check matrix in the form of a single row of circulant blocks, which will be of interest in the following chapters. Finally, we describe how to achieve efficient encoding algorithms based on fast polynomial multiplication and vector-circulant matrix products.

Keywords

Quasi-cyclic codes Circulant matrices Generator matrix Parity-check matrix Polynomial representation Fast vector-by-circulant-matrix product 

References

  1. 1.
    Townsend R, Weldon JE (1967) Self-orthogonal quasi-cyclic codes. IEEE Trans Inform Theory 13(2):183–195CrossRefzbMATHGoogle Scholar
  2. 2.
    Karlin M (1969) New binary coding results by circulants. IEEE Trans Inform Theory 15(1):81–92Google Scholar
  3. 3.
    Chen CL, Peterson WW, Weldon EJ Jr (1969) Some results on quasi-cyclic codes. Inform Contr 15:407–423CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Peterson WW, Weldon EJ (1972) Error-Correcting Codes, 2nd edn. MIT Press, CambridgezbMATHGoogle Scholar
  5. 5.
    MacWilliams F (1971) Orthogonal circulant matrices over finite fields, and how to find them. J Comb Theory Series A 10:1–17CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    MacWilliams FJ, Sloane NJA (1977) The theory of error-correcting codes. I and II. North-Holland Publishing Co, AmsterdamGoogle Scholar
  7. 7.
    Andrews K, Dolinar S, Thorpe J (2005) Encoders for block-circulant LDPC codes. In: Proceedings of IEEE international symposium on information theory ISIT, Adelaide, Australia, pp 2300–2304Google Scholar
  8. 8.
    Baldi M, Bodrato M, Chiaraluce F (2008) A new analysis of the McEliece cryptosystem based on QC-LDPC codes. In: Security and cryptography for networks. Lecture notes in computer science, vol 5229. Springer, Berlin, pp 246–262Google Scholar
  9. 9.
    Baldi M, Bianchi M, Chiaraluce F (2013) Security and complexity of the mceliece cryptosystem based on qc-ldpc codes. IET Inf Secur 7(3):212–220CrossRefGoogle Scholar
  10. 10.
    Karatsuba AA, Ofman Y (1963) Multiplication of multidigit numbers on automata. Sov Phys Dokl 7:595–596Google Scholar
  11. 11.
    Toom AL (1963) The complexity of a scheme of functional elements realizing the multiplication of integers. Sov Math Dokl 3:714–716Google Scholar
  12. 12.
    Cook SA (1966) On the minimum computation time of functions. PhD thesis, Department of Mathematics, Harvard UniversityGoogle Scholar
  13. 13.
    Bodrato M, Zanoni A (2007) Integer and polynomial multiplication: towards optimal Toom-Cook matrices. In: Brown CW (ed) Proceedings of the ISSAC 2007 conference, ACM press, pp 17–24Google Scholar
  14. 14.
    Cantor DG (1989) On arithmetical algorithms over finite fields. J Comb Theor A 50:285–300CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Schönhage A (1977) Schnelle multiplikation von polynomen über körpern der charakteristik 2. Acta Informatica 7:395–398CrossRefzbMATHGoogle Scholar
  16. 16.
    Brent RP, Gaudry P, Thom E, Zimmermann P (2008) Faster multiplication in GF(2)[x]. In: van der Poorten AJ, Stein A (eds) Proceedings of the eighth algorithmic number theory symposium (ANTS-VIII), Banff, Canada. Lecture notes in computer science, vol 5011. Springer, Berlin, pp 153–166Google Scholar
  17. 17.
    Bodrato M (2007) Towards optimal Toom-Cook multiplication for univariate and multivariate polynomials in characteristic 2 and 0. In: Carlet C, Sunar B (eds) WAIFI 2007 proceedings, Madrin, Spain. Lecture notes in computer science, vol 4547. Springer, Berlin, pp 116–133Google Scholar
  18. 18.
    Winograd S (1980) Arithmetic complexity of computations, CBMS-NSF regional conference series in mathematics, vol 33. SIAMGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.DIIUniversitá Politecnica delle MarcheAnconaItaly

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