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Keywords:

number systems, radix representation, complete digit sets, digit sets, SISTA, Science & Technology, Physical Sciences, Mathematics, DIGIT SETS, 0101 Pure Mathematics, General Mathematics, 4904 Pure mathematics

Abstract:

For an expanding matrix H epsilon Z(kxk), a subset W subset of Z(k) is called a complete digit set, if all points of the integer lattice Z(k) can be uniquely represented as a finite sum x = Sigma(N(x))(i=0) H(i)r(i), with r(i) epsilon W and N(x) epsilon N. We present a necessary and sufficient condition for the existence of a complete digit set in case vertical bar det(H)vertical bar = 2, implying that W is a binary complete digit set. This allows a characterization of the binary number systems (H, W) in Z(k). It is shown that, when H has a complete digit set, all its complete digit sets form a finitely generated Abelian group. Complete lists are given for dimension k = 1 to 6. (c) 2005 Elsevier Inc. All rights reserved.