Construction and Secrecy Gain of Formally Unimodular Lattices in Odd Dimensions

Abstract

In contrast to binary codes, odd-length self-dual codes exist over the integers modulo 4. Lately, the use of lattices constructed from codes over ℤ4 to guarantee secure communication in a Gaussian wiretap channel was proposed and shown to exceed the performance of lattices from binary codes. This performance is measured regarding the secrecy gain, a criterion that depends on a lattice’s volume and theta series. Formally unimodular lattices, i.e., lattices with the same theta series as their dual, have presented promising results with respect to the secrecy gain. While previous contributions in the literature were mainly focused on even-dimensional lattices, this paper addresses the secrecy gain of odd-dimensional formally unimodular lattices obtained from codes over ℤ4, together with a novel construction of such codes.