Abstract
We present a method of constructing free self-dual codes over \({\mathbb {Z}}_8\) and \({\mathbb {Z}}_{16}\) which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over \({\mathbb {Z}}_{2^m}\) can be found from a binary (extremal or optimal) Type II code for any positive integer \(m \ge 2\). We find explicit algorithms for construction of self-dual codes over \({\mathbb {Z}}_8\) and \({\mathbb {Z}}_{16}\). Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over \({\mathbb {Z}}_{2^m}\). By using our explicit algorithms, we construct extremal free self-dual codes over \({\mathbb {Z}}_8\) and \({\mathbb {Z}}_{16}\) up to lengths 40.
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Acknowledgments
Due to the referee’s helpful comments, we added Lemma 3.2 and Theorem 3.3. We express our gratitude to the referee. Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (2014-002731).
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Communicated by V. D. Tonchev.
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Kim, B., Lee, Y. Construction of extremal self-dual codes over \({\mathbb {Z}}_{8}\) and \({\mathbb {Z}}_{16}\) . Des. Codes Cryptogr. 81, 239–257 (2016). https://doi.org/10.1007/s10623-015-0137-8
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DOI: https://doi.org/10.1007/s10623-015-0137-8