Codes Based on Complete Graphs

Dieter Jungnickel, Marialuisa J. De Resmini and Scott A. Vanstone
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Dieter Jungnickel: Universität Augsburg, Lehrstuhl für Angewandte Mathematik II
Marialuisa J. De Resmini: Università di Roma “La Sapienza”, Dipartimento di Matematica
Scott A. Vanstone: University of Waterloo, Dept. of Combinatorics and Optimization

A chapter in Designs and Finite Geometries, 1996, pp 159-165 from Springer

Abstract: Abstract We consider the problem of embedding the even graphical code based on the complete graph on n vertices into a shortening of a Hamming code of length 2 m - 1, where m = h(n) should be as small as possible. As it turns out, this problem is equivalent to the existence problem for optimal codes with minimum distance 5, and optimal embeddings can 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 + 1 and to narrow down the possibilities in general to two or three conceivable values.

Date: 1996
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DOI: 10.1007/978-1-4613-1395-3_11

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