Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes

Steven T. Dougherty; Cristina Fernández-Córdoba

doi:10.3934/amc.2011.5.571

Article Contents

2011, Volume 5, Issue 4: 571-588. Doi: 10.3934/amc.2011.5.571

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Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes

Steven T. Dougherty^1, and
Cristina Fernández-Córdoba^2,

1.
Department of Mathematics, University of Scranton, Scranton, PA 18510
2.
Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra

Received: February 28, 2010

Revised: September 30, 2011

Published: October 2011

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Abstract

The generalized Gray map is defined for codes over $\mathbb{Z}_{2^k}$. We give bounds for the dimension of the kernel and the rank of the image of a code over $\mathbb{Z}_{2^k}$ with a given type and show that there exists such a code for each dimension in the interval for the kernel. We determine when the Gray image of a code over $\mathbb{Z}_{2^k}$ generates a linear self-dual code and give families of codes whose image generate binary self-dual codes. We investigate the Gray image of quaternary self-dual codes and examine when the Gray image of a self-dual code over $\mathbb{Z}_4$ is a binary self-dual code.

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Mathematics Subject Classification: Primary: 94B25, 94B60.

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