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On Perfect Codes and Related Concepts

  • Rudolf AhlswedeEmail author
Chapter
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 15)

Abstract

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere-packing bound) is introduced. This was motivated by the “code-anticode” bound of Delsarte in distance regular graphs. This bound in conjunction with the complete solutions of diametric problems in the Hamming graph \(\mathcal H_q(n)\) and the Johnson graph J(nk) gives a sharpening of the sphere-packing bound. Some necessary conditions for the existence of diameter perfect codes are given. In the Hamming graph all diameter perfect codes over alphabets of prime power size are characterized. The problem of tiling of the vertex set of J(nk) with caps (and maximal anticodes) is also examined.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BielefeldGermany

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