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

Abstract

We show how cyclic ( f, σ, δ)-codes over finite rings canonically induce a Zlattice in 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 σ -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.

Nonassociative Differential Extensions of Characteristic $$\varvec{p}$$ p

“…Let δ have minimum polynomial Remark 21. Petit's construction of nonassociative algebras S f can be generalized to the setting where f ∈ S[t; δ] is a monic polynomial and S any unital associative ring [14]. Therefore some of the results above also hold for nonassociative algebras obtained by employing f (t) = t p − t − d ∈ S[t; δ] if δ satisfies the polynomial identity δ p = δ as before.…”

On the structure of repeated-root polycyclic codes over local rings

“…Then, they generalized the (σ, δ)-polycyclic codes over fields into T M -invariant linear codes. As applications, in [28,29], (σ, δ)-polycyclic codes were used to establish good quantum codes, and in [30], these codes were applied to construct Z-lattices in R N . It is remarkable that (σ, δ)-polycyclic codes have been described by different terminologies in the above references, such as "(σ, δ)-codes", "( f, σ, δ)-codes", "(M, σ, δ)-codes", and "skew generalized cyclic codes".…”

Division algebras and MRD codes from skew polynomials

“…We construct these MRD codes using skew polynomials. Skew polynomials have been successfully used in constructions of both division algebras (mostly semifields) and linear codes [2,4,5,13,[26][27][28], in particular building space-time block codes (STBCs) [29] and MRD codes [33,34].…”