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.

Parts of the paper have been rewritten and some proofs included, Section 6 has been dropped

References in corpus (4)

• The nonassociative algebras used to build fast-decodable space-time block codes
• Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,\sigma ,\delta )$-codes
• Self-dual cyclic codes over finite chain rings
• A note on linear codes and nonassociative algebras obtained from skew-polynomial rings

Cited by in corpus (3)

• Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,\sigma ,\delta )$-codes
• How a nonassociative algebra reflects the properties of a skew polynomial
• Quotients of orders in algebras obtained from skew polynomials with applications to coding theory