Abstract
The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K 4 on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m 3/36+7m 2/24+11m/12+1−(m mod 2)/8−(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).
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15 October 2017
The acknowledgment (footnote to the title of the paper) should read as follows: The research was carried out at the expense of the Russian Science Foundation, project no. 14-11-00555.
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Original Russian Text © E.A. Bespalov, D.S. Krotov, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 2, pp. 40–59.
Supported in part by the Russian Foundation for Basic Research, project no. 14-11-00555.
An erratum to this article is available at https://doi.org/10.1134/S0032946017030139.
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Bespalov, E.A., Krotov, D.S. MDS codes in Doob graphs. Probl Inf Transm 53, 136–154 (2017). https://doi.org/10.1134/S003294601702003X
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DOI: https://doi.org/10.1134/S003294601702003X