Advances in Mathematics of Communications (AMC)

Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order

Pages: 57 - 90, Volume 7, Issue 1, February 2013      doi:10.3934/amc.2013.7.57

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W. Cary Huffman - Department of Mathematics and Statistics, Loyola University, Chicago, IL 60660, United States (email)

Abstract: Additive codes over $\mathbb{F}_4$ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic $p$ are connected to codes over $\mathbb{F}_{p^2}$ that are $\mathbb{F}_p$-linear in [30]. These codes that arise as connections with quantum codes are self-orthogonal under a particular inner product. We study a further generalization to codes termed $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are self-dual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case $q=3$ and $t=2$.

Keywords:  Additive codes, self-dual codes, code automorphisms.
Mathematics Subject Classification:  94B15.

Received: June 2012;      Revised: August 2012;      Available Online: January 2013.