| Abstract: |
| Since the pioneering work of Claude Shannon, lattices, namely, discrete subgroups of $\mathbb R^n$, have always occupied a prominent position in information theory and communications as they are used in modulation design. Lattices have also played an important role in cryptography, more specifically, in connection with post-quantum cryptographic systems. For communication purposes, one of the most relevant parameters of a lattice is its sphere-packing density. Interesting lattice packings have been constructed from error-correcting codes, groups, number fields, and elliptic curves. In this talk laminated lattices of full diversity in odd dimensions $d$ in $[3..15]$ will be constructed using a combination of Abelian number fields and error-correcting codes. More specifically, we will first consider a totally real Abelian field $F$ of degree $d$ and then obtain a $d$-dimensional lattice from its ring of integers, $\frak O_F$, via the canonical embedding (Minkowski homomorphism). Secondly, a $\mathbb Z$-submodule $\EuScript M$ of $\frak O_F$ defined by the parity-check matrix of a Reed-Solomon code is considered. We show that the image of $\EuScript M$ under the canonical embedding is isometric to $\Lambda_d$, the $d$-dimensional laminated lattice. |
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