Codes Based on Complete Graphs

  • Authors:

  • Dieter Jungnickel;Marialuisa J. De Resmini;Scott A. Vanstone

  • Affiliations:

  • Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, D-86135 Augsburg, Germany;Dipartimento di Matematica, Università di Roma ``La Sapienza'', 2, Piazzale Aldo Moro, I-00185 Roma, Italy;Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont., N2L 3G1, Canada

  • Venue:
  • Designs, Codes and Cryptography - Special issue dedicated to Hanfried Lenz
  • Year:
  • 1996

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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 2^m - 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 2^k +1and to narrow down the possibilities in general to two or threeconceivable values.