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How does a porcupine separate its quills?

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Sets of unit vectors inN-dimensional Euclidean vector space whose constituent vectors are separated one from another by at least a fixed distanced, prescribed once for all and independent ofN, are of interest in theory and practice; they have fondly been called "porcupine codes." Although an elegant constructive proof of Gilbert shows that the number of vectors in a porcupine code (of givend) can increase exponentially withN, no systematic method is yet known for generating porcupine codes of this cardinality. Corresponding to a collection ofMvectors, we can partition the space into maximum-likelihood regions, thejth of which consists of those vectors that lie closer to thejth than to any other element of the collection. Each maximum-likelihood region is bounded by at most(M - 1)hyperplanes, and we denote byKthe total number of these bounding hyperplanes. Collections for whichKis small may be expected to have greater symmetry than those for whichKis large. In this paper we show that, for porcupine codes,K geq (M/2)^{1/s}, withsdepending only ond, 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 withN.

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Information Theory, IEEE Transactions on  (Volume:17 ,  Issue: 2 )