Advances in Mathematics of Communications (AMC)

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

Pages: 571 - 588, Volume 5, Issue 4, November 2011      doi:10.3934/amc.2011.5.571

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Steven T. Dougherty - Department of Mathematics, University of Scranton, Scranton, PA 18510, United States (email)
Cristina Fernández-Córdoba - Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain (email)

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.

Keywords:  $\mathbb{Z}_{2^k}$-codes, self-duality, Gray map.
Mathematics Subject Classification:  Primary: 94B25, 94B60.

Received: March 2010;      Revised: October 2011;      Available Online: November 2011.