Secrecy Gain of Formally Unimodular Lattices From Codes Over the Integers Modulo 4

Recently, a design criterion depending on a lattice's volume and theta series, called the secrecy gain, was proposed to quantify the secrecy-goodness of the applied lattice code for the Gaussian wiretap channel. To address the secrecy gain of Construction A4 lattices from formally self-dual <tex-math notation="LaTeX">Z₄ </tex-math> -linear codes, i.e., codes for which the symmetrized weight enumerator (swe) coincides with the swe of its dual, we present new constructions of <tex-math notation="LaTeX">Z₄ </tex-math> -linear codes which are formally self-dual with respect to the swe. For even lengths, formally self-dual <tex-math notation="LaTeX">Z₄ </tex-math> -linear codes are constructed from nested binary codes and double circulant matrices. For odd lengths, a novel construction called odd extension from double circulant codes is proposed. Moreover, the concepts of Type I/II formally self-dual codes/unimodular lattices are introduced. Next, we derive the theta series of the formally unimodular lattices obtained by Construction A4 from formally self-dual <tex-math notation="LaTeX">Z₄ </tex-math> -linear codes and describe a universal approach to determine their secrecy gains. The secrecy gain of Construction A4 formally unimodular lattices obtained from formally self-dual <tex-math notation="LaTeX">Z₄ </tex-math> -linear codes is investigated, both for even and odd dimensions. Numerical evidence shows that for some parameters, Construction A4 lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Results concerning the flatness factor, another security criterion widely considered in the Gaussian wiretap channel, are also discussed.