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).
Problems of Information Transmission – Springer Journals
Published: Jul 13, 2017
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