Abstract
We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited.
Highlights
We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields
Let bα : OKN × OKN → R be the symmetric bilinear form defined by bα(x, y) = TrK/Q(αxiyi) i=1 where α ∈ K ∩ R and yi denotes the complex conjugate of yi if K is CM
We propose two approaches to discuss the modularity of lattices obtained via the above method
Summary
Let K be a Galois number field of degree n which is either totally real or a CM field. In [1], the quadratic fields Q( −7) with p = (2), Q(i) with p = (2) and Q(ζ3) with p = (2) or p = (3), as well as totally definite quaternion algebras ramified at either 2 or 3 with p = (2), were used to construct modular lattices from self-dual codes. Interesting examples are found – new constructions of known extremal lattices, modular lattices with large minimal norm – and numerical evidence gives new insight on the behaviour of the secrecy gain
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