Abstract
In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to \(R_k\). Using this method we construct self-dual codes over \(R_3\) of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have \(\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41\) and 52 in \(W_{64,2}\) and they all have automorphism groups of size 8.
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The authors would like to thank the anonymous referees for their valuable remarks and suggestions that improved the presentation of the paper considerably.
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Communicated by J.-L. Kim.
This paper was prepared and submitted before [15] in which it was cited, but the latter got published before the current paper.
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Karadeniz, S., Yildiz, B. New extremal binary self-dual codes of length 64 from \(R_3\)-lifts of the extended binary Hamming code. Des. Codes Cryptogr. 74, 673–680 (2015). https://doi.org/10.1007/s10623-013-9884-6
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DOI: https://doi.org/10.1007/s10623-013-9884-6