Abstract
By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs with the same parameters as the above 5-designs. We give some parameters of a self-orthogonal 5-design whose existence is equivalent to that of an extremal doubly even self-dual code of length 24m for \(m=3,4,5,6\). If \(m \in \{1,\ldots ,6\}\), \(k \in \{m+1,\ldots ,5m-1\}\) and \((m,k) \ne (6,18)\), then it is shown that an extremal doubly even self-dual code of length 24m is generated by codewords of weight \(4k\).
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Acknowledgments
The author would like to thank Tsuyoshi Miezaki for verifying the calculations in the proofs of Propositions 6 and 7, independently. This work is supported by JSPS KAKENHI Grant Number 23340021. This work was partially carried out at Yamagata University.
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Communicated by J. D. Key.
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Harada, M. On a 5-design related to a putative extremal doubly even self-dual code of length a multiple of 24. Des. Codes Cryptogr. 76, 373–384 (2015). https://doi.org/10.1007/s10623-014-9963-3
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DOI: https://doi.org/10.1007/s10623-014-9963-3