Lattices from maximal orders into quaternion algebras

Lattices Associated with Octonion Algebras

“…The proof regarding the reduced norm follows the same way. So, by Proposition 5 the set O generated by the basis C is an octonion order in C n for all n. Now, we consider again the field K n for some n ≥ 5 and the basis C of C n as in (11). Associated to this basis we have det (B) = 2 12 and we want to choose an appropriate α = (2 − η n ) m for some m ≥ 0 such that det (G) = 2 k2 n , k integer.…”

“…, a number field of degree 2, and let C 4 = (−1, −1, −1) K 4 the octonion algebra over this field K 4 with basis constructed as in (11). Now, taking {1, √ 2}, a basis of the ring of integers o K , we can construct a lattice Λ = (O, 2 − √ 2) with dimension 16.…”

“…Maximal quaternion orders have been proposed in the context of space-time block codes in [9] and complex codes constructions based on cyclic division algebras are proposed in [10]. More recently, the E 8 -lattice was constructed over an imaginary quadratic field [11]. Codewords are usually built over the complex field.…”

Lattices from maximal quaternion orders over totally real number fields

“…whose rows give us a new basis of the lattice Λ with non-vanishing elements which improve the diversity of Λ. Hence, expanding M 1 into a 8 × 8 matrix we have the matrix φ 1 as in (3). Multiplying φ 1 by φ 2 we have a generator matrix M as in (4) of the lattice Λ with non-vanishing elements.…”

“…Maximal orders have been proposed in the context of space-time block codes in [7] and complex codes constructions based on cyclic division algebras are proposed in [16]. More recently, the E 8 -lattice was constructed using quaternion algebras over an imaginary quadratic field [3]. Codewords are usually (in narrow band systems) built over the complex field.…”