On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

W. Cary Huffman

doi:10.3934/amc.2013.7.349

Article Contents

2013, Volume 7, Issue 3: 349-378. Doi: 10.3934/amc.2013.7.349

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On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

W. Cary Huffman^1,

1.
Department of Mathematics and Statistics, Loyola University, Chicago, IL 60660

Received: February 28, 2013

Published: July 2013

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Abstract

In [7], self-orthogonal additive codes over $\mathbb{F}_4$ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. We examine a number of classical results from the theory of $\mathbb{F}_q$-linear codes, and see how they must be modified to give analogous results for $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.

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Mathematics Subject Classification: Primary: 94B60, 94B05; Secondary: 94B27.

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