Duality between Packings and Coverings of the Hamming Space

Details Duality between Packings and Coverings of the Hamming Space Duality between Packings and Coverings of the Hamming Space

2005-07-06

arxiv.org/abs/cs/0507015v1

Computer Science

Information Theory

5 pages

Scientific paper

We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all codes are good coverings, then at most a vanishing fraction of codes are good packings. We also show that any specific maximal binary code is either a good packing or a good covering, in a certain well-defined sense.

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