Abstract
In Etzion et al. (IEEE Trans Inform Theory 64(4):2398–2409, 2018), introduced metrics on \(\mathbb {F}_2^n\) based on directed graphs on n vertices, and in Hyun et al. (IEEE Trans Inform Theory 65(8):4664–4672, 2019), the authors introduced weighted poset metrics on \(\mathbb {F}_2^n\) which may be considered as an algebraic version of directed graph metric. They also classify weighted posets and directed graphs admitting the extended Hamming code of length 8 to be a 2-perfect code. In this paper, we will continue the classification problem further. It will be shown that every weighted poset or directed graph which admits extended Hamming code is completely determined by its structure vector. Let s be the length of the structure vector. In Sect. 3 (resp.4), we will classify structure vectors of weighted posets (resp. directed graphs) which admit the extended Hamming code \({\widetilde{\mathcal {H}}}_m, m \ge 2\) to be a 2-perfect code when \(s=3,4,5\) (resp. \(s=3,4\)). As a consequence our classification, we will classify weighted posets and directed graphs which admit the extended Hamming code of length 16 to be a 2-perfect code.
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Acknowledgements
This research was supported by the BK21 FOUR (Fostering Outstanding Universities for Research) funded by the Minister of Education (MOE, Korea) and National Research Foundation of Korea(NRF) (4.0021515).
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”.
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Kim, H.K., Kwon, J. Classification of weighted posets and digraphs admitting the extended Hamming code to be a perfect code. Des. Codes Cryptogr. 90, 2249–2269 (2022). https://doi.org/10.1007/s10623-022-01066-8
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DOI: https://doi.org/10.1007/s10623-022-01066-8