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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the existence of extremal Type II $ \mathbb{Z}_{2k}$-codes


Authors: Masaaki Harada and Tsuyoshi Miezaki
Journal: Math. Comp. 83 (2014), 1427-1446
MSC (2010): Primary 94B05; Secondary 11H71, 11F11
Published electronically: July 24, 2013
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Abstract: For lengths $ 8$, $ 16$, and $ 24$, it is known that there is an extremal Type II $ \mathbb{Z}_{2k}$-code for every positive integer $ k$. In this paper, we show that there is an extremal Type II $ \mathbb{Z}_{2k}$-code of lengths $ 32,40,48,56$, and $ 64$ for every positive integer $ k$. For length $ 72$, it is also shown that there is an extremal Type II $ \mathbb{Z}_{4k}$-code for every positive integer $ k$ with $ k \ge 2$.


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Additional Information

Masaaki Harada
Affiliation: Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan
Email: mharada@sci.kj.yamagata-u.ac.jp

Tsuyoshi Miezaki
Affiliation: Department of Mathematics, Oita National College of Technology, 1666 Oaza-Maki, Oita, 870–0152, Japan
Address at time of publication: Faculty of Education, Art and Science, Yamagata University, Yamagata 990-8560, Japan
Email: miezaki@e.yamagata-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02750-0
PII: S 0025-5718(2013)02750-0
Keywords: Self-dual code, unimodular lattice, $k$-frame, theta series
Received by editor(s): May 31, 2012
Received by editor(s) in revised form: July 24, 2012
Published electronically: July 24, 2013
Dedicated: In memory of Boris Venkov
Article copyright: © Copyright 2013 American Mathematical Society