On skew polynomial codes and lattices from quotients of cyclic division algebras

Jérôme  Ducoat; Frédérique Oggier

doi:10.3934/amc.2016.10.79

Article Contents

2016, Volume 10, Issue 1: 79-94. Doi: 10.3934/amc.2016.10.79

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On skew polynomial codes and lattices from quotients of cyclic division algebras

Jérôme Ducoat^1, and
Frédérique Oggier^1,

1.
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University

Received: December 2014

Revised: August 2015

Published: February 2016

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Abstract

We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak{p}\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak{p}$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak{p}\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.

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Mathematics Subject Classification: Primary: 11S45; Secondary: 11T71, 94B40.

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