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Codes Based on Complete Graphs

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Abstract

We consider the problem of embedding the even graphicalcode based on the complete graph on n vertices intoa shortening of a Hamming code of length 2m - 1,where m=h(n) should be as small as possible. Asit turns out, this problem is equivalent to the existence problemfor optimal codes with minimum distance 5, and optimal embeddingscan always be realized as graphical codes based on K n.As a consequence, we are able to determine h(n)exactly for all n of the form 2k +1 and to narrow down the possibilities in general to two or threeconceivable values.

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Authors and Affiliations

  1. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, D-86135, Augsburg, Germany

    Dieter Jungnickel

  2. Dipartimento di Matematica, Universitàdi Roma ``La Sapienza'', 2, Piazzale Aldo Moro, I-00185, Roma, Italy

    Marialuisa J. de Resmini & Scott A. Vanstone

Authors
  1. Dieter Jungnickel
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  2. Marialuisa J. de Resmini
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  3. Scott A. Vanstone
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Jungnickel, D., Resmini, M.J.d. & Vanstone, S.A. Codes Based on Complete Graphs. Designs, Codes and Cryptography 8, 159–165 (1996). https://doi.org/10.1023/A:1018089026819

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  • DOI: https://doi.org/10.1023/A:1018089026819

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