Using cyclic $(f,\sigma)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices

TL;DR

This paper introduces a novel method for constructing $Z_p$- and $F_q[igl[Tigr]igr]$-lattices using cyclic $(f,\sigma)$-codes over finite chain rings, extending classical lattice construction techniques to nonassociative algebraic structures.

Contribution

It generalizes the classical Construction A to nonassociative settings, enabling the creation of new lattice and rank-metric codes from cyclic $(f,\sigma)$-codes over finite chain rings.

Findings

Constructed $Z_p$-lattice codes using nonassociative cyclic algebras.

Developed linear MRD codes as $Z_p$-lattice codes via algebraic multiplication.

Extended lattice construction methods to nonassociative algebraic frameworks.

Abstract

We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,\sigma)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of $p$-adic fields. This means we focus on $\sigma$-constacyclic codes to obtain $\mathbb{Z}_p$-lattices, hence $\mathbb{Z}_p$-lattice codes. We construct linear maximum rank distance (MRD) codes that are $\mathbb{Z}_p$-lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible applications of our constructions include post-quantum cryptography involving $p$-adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or $p$-adic coset coding, in particular wire-tap coding.