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How to obtain lattices from \((f,\sigma,\delta)\)-codes via a generalization of construction A. (English) Zbl 1436.94113

Summary: 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 J. Ducoat and F. Oggier [Adv. Math. Commun. 10, No. 1, 79-94 (2016; Zbl 1352.94093)], 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.

MSC:

94B05 Linear codes (general theory)
94B40 Arithmetic codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
17A35 Nonassociative division algebras

Citations:

Zbl 1352.94093

Software:

Plural
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References:

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