How to obtain lattices from (f,sigma,delta)-codes via a generalization of Construction A

Abstract: We show how cyclic $(f,\sigma,\delta)$-codes over finite rings canonically induce a $\mathbb{Z}$-lattice in $\mathbb{R}^N$ by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial $f$. This construction generalizes the one using certain $\sigma$-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by $f$, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases.