How does a porcupine separate its quills?

Abstract

Sets of unit vectors in N -dimensional Euclidean vector space whose constituent vectors are separated one from another by at least a fixed distance d , prescribed once for all and independent of N , are of interest in theory and practice; they have fondly been called codes. Although an elegant constructive proof of Gilbert shows that the number of vectors in a porcupine code (of given d ) can increase exponentially with N , no systematic method is yet known for generating porcupine codes of this cardinality. Corresponding to a collection of M vectors, we can partition the space into maximum-likelihood regions, the j th of which consists of those vectors that lie closer to the j th than to any other element of the collection. Each maximum-likelihood region is bounded by at most (M - 1) hyperplanes, and we denote by K the total number of these bounding hyperplanes. Collections for which K is small may be expected to have greater symmetry than those for which K is large. In this paper we show that, for porcupine codes, K \geq (M/2)^{1/s} , with s depending only on d , the minimum separation of the code vectors. Hence, for the number of vectors of a porcupine code to increase exponentially with dimension, the number of separating hyperplanes must do so as well. We conclude with, an application to the permutation codes introduced by Slepian, showing that the number of vectors of a porcupine code which is of permutation-modulation type can not increase exponentially with N .