On binary codes of order 3

  • B. Courteau
  • A. Montpetit
Extended Abstracts
Part of the Lecture Notes in Computer Science book series (LNCS, volume 357)


We present a class of binary (not necessarily linear) codes containing perfect codes, Preparata codes, BCH codes of lengtht 22m+1-1, Hadamard code of length 11, l-error-correcting uniformly packed codes and also some other classes presenting remarkable regularity properties. The points of a code C of order 3 are scattered in the ambiant Hamming space so that for any point x the number of paths of length 3 joining x to the code only depends on the fact that the Hamming distance d(x, C) from x to C is 0, 1 or is greater than 1.

In the linear case the set of columns of a parity check matrix of a code of order 3 is a triple-sum-set [5], which is a natural extension of partial difference sets [7, 15, 2] and the orthogonal of such a code admits at most three non-zero weights.

The aim of this communication is to introduce and characterize codes of order 3 and of order 3-star in the binary case and to give some examples. The general q-any case is treated in


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  1. [1]
    A.R. Calderbank and J.-M. Goethals, Three-Weight Codes and Association Schemes, Philips J. Res. 39 (1984) 143–152.Google Scholar
  2. [2]
    P. Camion, Difference Sets in Elementary Abelian Groups (Les Presses de l'Université de Montréal, Montréal, 1979).Google Scholar
  3. [3.
    P. Camion, B. Courteau, P. Delsarte, On r-partition Designs in Hamming Spaces, Rapport de recherche no 626. INRIA, Rocquencourt, 78153 Le Chesnay, France, février 1987.Google Scholar
  4. [4]
    B. Courteau, A. Montpetit, On a class of codes admitting at most three non-zero dual distances, submitted to Discrete Math.Google Scholar
  5. [5]
    B. Courteau and J. Wolfmann, On Triple-sum-sets and two or three weights codes, Discrete Math. 50 (1984) 179–191.Google Scholar
  6. [6]
    B. Courteau, G. Fournier, R. Fournier, A Characterization of N-weights projective codes, IEEE Trans. Inform. Theory 27 (1981) 808–812.Google Scholar
  7. [7]
    I.M. Chakravarti and K.V. Suryanarayana, Partial difference sets and partially balanced weighting desings for calibration and tournaments, J. Combinatorial Theory (A) 13 (1972) 426–431.Google Scholar
  8. [8]
    P. Delsarte, Four Fundamental Parameters of a Code and Their Combinatorial Significance, Inform. and control 23 (1973) 407–438.Google Scholar
  9. [9]
    P. Dembowski, Finite Geometrics (Springer-Verlag, New York, 1968).Google Scholar
  10. [10]
    J.-M. Goethals and H.C.A. Van Tilborg, Uniformly packed codes, Philips Research Reports 30 (1975) 9–36.Google Scholar
  11. [12]
    H.C.A. Van Tilborg, Uniformly packed codes, Thesis, Tech. Univ. Eindhoven, 1976.Google Scholar
  12. [13]
    J. Wolfmann, Codes projectifs à deux ou trois poids associés aux hyperquadriques d'une géométrie finie, Discrete Math. 13 (1975), 185–211.Google Scholar
  13. [14]
    J. Wolfmann, Codes projectifs à deux poids, ‘caps’ complets et ensembles de différences, J. Combinatorial Theory (A) 23 (1977).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • B. Courteau
    • 1
  • A. Montpetit
    • 1
  1. 1.Département de mathématiques et d'informatiqueUniversité de SherbrookeSherbrookeCanada

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