Binary number systems for

✍ Scribed by A. Barbé; F. von Haeseler

• Book ID: 104024603
• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 196 KB
• Volume: 117
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.05.006

✦ Synopsis

For an expanding matrix H ∈ Z k×k , a subset W ⊂ 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 = N(x) i=0 H i r i , with r i ∈ W and N(x) ∈ N. We present a necessary and sufficient condition for the existence of a complete digit set in case |det(H )| = 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.