Similar Sublattices of Planar Lattices

Abstract

Abstract. The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Multiple description lattice vector quantization using multiple A4 quantizers

“…The similar sub lattices of A 4 are in one-to-one relation to those of the lattice L. If p ∈ I then pLp ⊂ L [7]. This means that all similar sub lattices (SSL) of the lattice L are images of L under orientation preserving mappings of the form x → αpxp, with p ∈ I − primitive and α ∈ Q [7].…”

“…It is difficult to find the similar lattices of A 4 using the description of A 4 as a lattice in a 4-dimensional hyperplane of R 5 [7,8]. Therefore the description in R 4 space is usually used:…”

“…This means that all similar sub lattices (SSL) of the lattice L are images of L under orientation preserving mappings of the form x → αpxp, with p ∈ I − primitive and α ∈ Q [7].…”

“…This description enables us to employ the quaternion algebra, based on the inclusion of a root system of type A 4 within a type of H 4 [7]. The Hamiltonian quaternion algebra H(K) is defined…”

“…The choice of L among many realizations of the root lattice A 4 is motivated by the observation that L is a subset of the so-called icosian ring I [7]. The icosian group is a multiplicative group of order 120 consisting of the quaternions…”