Odd and even hamming spheres also have minimum boundary

Janos Körner, Victor K. Wei
1984 Discrete Mathematics  
Combinatorial problems with a geometric flavor arise if the set of all binary sequences of a fixed len@h n, is provided with the Hamming distance. The Hamming dkcunce of any two binary sequences is the number of positions in which they differ. The (outer) boundary of a set A of binary sequences is the set of ail sequences outside A that are at distance 1 Erom some sequence in A. Harper [6] proved that among all the sets of a prescribed volume, the 'sphere' has minimum boundary. We show that
more » ... g all the sets in which no pair of sequences havz distance 1, the set of all the sequences with an even (odd) number of k's in a Hamn'ing 'sphere' has the same minimizing property. Some related results are obtained. Sets with more general extremal properties of this kind yield good error-correcting codes for multi-terminal channels.
doi:10.1016/0012-365x(84)90068-2 fatcat:tzqlig6m7rbwxocwfhvobgsd4y