February  2007, 1(1): 93-97. doi: 10.3934/amc.2007.1.93

Duality between packings and coverings of the Hamming space

1. 

Département Informatique, Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, 75634 Paris, France

2. 

Department of Electrical and Computer Engineering, Department of Computer Science and Engineering, Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA92093, United States

Received  May 2006 Revised  June 2006 Published  January 2007

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.
Citation: Gérard Cohen, Alexander Vardy. Duality between packings and coverings of the Hamming space. Advances in Mathematics of Communications, 2007, 1 (1) : 93-97. doi: 10.3934/amc.2007.1.93
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