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

We construct $$\mathbb {Z}_p$$ Z p -lattices and $$\mathbb {F}_q[\![t]\!]$$ F q [ [ t ] ] -lattices from cyclic $$(f,\sigma )$$ ( f , σ ) -codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen and not necessarily associative algebras. This approach generalizes the classical Construction A that obtains $$\mathbb {Z}$$ Z -lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use (associative or nonassociative) cyclic algebras that are defined over field extensions of p -adic fields. This means we focus on certain skew constacyclic codes over finite rings to obtain $$\mathbb {Z}_p$$ Z p -lattices, hence $$\mathbb {Z}_p$$ Z p -lattice codes. We construct linear maximum rank distance (MRD) codes that are $$\mathbb {Z}_p$$ Z p -lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible other applications of our constructions include cryptography involving p -adic lattices, e.g. learning with errors.