Complex Gaussian noise moments

Sets of unit vectors in N-dimensional Eucl idean vectorspace whose constituent vectors are separated one from another by atleast a fixed distance d, prescr ibed once for all and independent of N,are of interest in theory and practice; they have fondly been called“porcupine codes.” Although an elegant constructive proof of Gilbertshows that the number of vectors in a porcupine code (of given d) canincrease exponential ly with N, no systematic method is yet known forgenerat ing porcupine codes of this cardinality.Corresponding to a collection of M vectors, we can partition the spaceinto maximum-likel ihood regions, the jth of which consists of thosevectors that lie closer to thejth than to any other element of the collection.Each maximum-likel ihood region is bounded by at most (M 1) hyper-planes, and we denote by K the total number of these boundinghyperplanes. Collections for which K is small may be expected to havegreater symmetry than those for which K is large.In this paper we show that, for porcupine codes, K 2 (M/2)““, withs depending only on d, the minimum separat ion of the code vectors.Hence, for the number of vectors of a porcupine code to increase ex-ponentially with dimension, the number of separat ing hyperplanes mustdo so as well. W e conclude with, an application to the permutat ion codesintroduced by Slepian, showing that the number of vectors of a porcupinecode which is of permutat ion-modulat ion type can not increaseexponential ly with N. Manuscript received May 11, 1970.The author is with Bell Te lephone Laboratories, Inc., Murray Hill,N.J. 07974.