Designs, Codes and Cryptography

, Volume 45, Issue 1, pp 51–64 | Cite as

G-projectable and Λ-projectable binary linear block codes

  • Morteza Esmaeili
  • Morteza Hivadi


The binary [24,12,8] Golay code has projection O onto the quaternary [6,3,4] hexacode [9] and the [32,16,8] Reed-Muller code has projection E onto the quaternary self-dual [8,4,4] code [6]. Projection E was extended to projection G in [8]. In this paper we introduce a projection, to be called projection Λ, that covers projections O, E and G. We characterise G-projectable self-dual codes and Λ-projectable codes. Explicit methods for constructing codes having G and Λ projections are given and several so constructed codes that have best known optimal parameters are introduced.


Linear block code Projection on F_4 G-Projection Λ-Projection 

AMS Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amrani O and Be’ery Y (2001). Reed-Muller codes: Projections on GF(4) and multilevel construction. IEEE Trans Inform Theory 47: 2560–2565 MATHCrossRefGoogle Scholar
  2. Brouwer AE Linear code bound. Scholar
  3. Esmaeili M, Gulliver TA and Khandani AK (2003). On the Pless construction and ML Decoding of the (48,24,12) quadratic residue code. IEEE Trans Inform Theory 49: 1527–1535 MATHCrossRefGoogle Scholar
  4. Esmaeili M, Gulliver TA On the quaternary projection of binary linear block codes. ARS Combinatoria, Vol.~85, October~2007Google Scholar
  5. Gaborit P, Huffman WC, Kim J-L, Pless V (2001) On additive GF(4) codes. In: proceedings of the DIMACS workshop on codes and association schemes, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56, AMS Providence, RI, pp 135–149Google Scholar
  6. Gaborit P, Kim J-L and Pless V (2003). Decoding binary R(2,5) by hand. Discrete Math 264: 55–73 MATHCrossRefGoogle Scholar
  7. Kim J-L, Pless V (2002) Decoding some doubly-even self-dual [32,16,8] codes by hand. In: proceedings of a conference honouring Professor Dijen Ray-Chaudhuri on the occasion of his 65th birthday, Ohio State Univ. Math. Res. Inst. Publ. 10, Ohio State University, Columbus, OH, pp 165–178Google Scholar
  8. Kim J-L, Mellinger KE and Pless V (2003). Projections of binary linear codes onto larger fields. SIAM J Discrete Math 16(4): 591–603 MATHCrossRefGoogle Scholar
  9. Pless V (1986). Decoding the Golay codes. IEEE Trans Inform Theory 32: 561–566 MATHCrossRefGoogle Scholar
  10. Vardy A and Be’ery Y (1991). More efficient soft decoding of the Golay codes. IEEE Trans Inform Theory 37: 667–672 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

Personalised recommendations