Abstract
Consider an assignment of bits to the vertices of a connected graph \(\Gamma (V, E)\) with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length |V| on \(\Gamma \). In this paper we construct an infinite family of linear storage codes on triangle-free graphs with rates arbitrarily close to one.
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Acknowledgements
While we are finishing writing up this paper, we became aware of [1], in which the authors construct a family of storage codes of rate asymptotically one. Since both the construction in the current paper and the construction in [1] are generalizing the Hamming family, they look similar. However the methods used to compute the rates of storage codes are very different.
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Communicated by T. Feng
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Research partially supported by the National Natural Science Foundation of China grant 12071206, 12131011, 12150710510, and the Sino-German Mobility Programme M-0157.
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Huang, H., Xiang, Q. Construction of storage codes of rate approaching one on triangle-free graphs. Des. Codes Cryptogr. 91, 3901–3913 (2023). https://doi.org/10.1007/s10623-023-01278-6
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DOI: https://doi.org/10.1007/s10623-023-01278-6