ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 2, pp. 136–154.
Pleiades Publishing, Inc., 2017.
Original Russian Text
E.A. Bespalov, D.S. Krotov, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 2, pp. 40–59.
MDS Codes in Doob Graphs
E. A. Bespalov
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Received February 6, 2016; in ﬁnal form, December 4, 2016
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
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
/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).
We study MDS codes in Doob graphs with code distance d ≥ 3.
An MDS code with parameters (N,q
,d) in the Hamming graph H(N,q) is a code of cardinal-
with code distance d = N − k + 1. Studying MDS codes in Hamming graphs is an important
area in coding theory (see  for details). In the general case, the question on existence and clas-
siﬁcation of MDS codes with given parameters is an open problem. For the classiﬁcation of MDS
codes for small values of q (no greater than 8), see [2–4].
The case of q = 4 is special for Hamming graphs H(N,q), since only in this case the graph
H(N,4), N ≥ 2, is not deﬁned as a distance-regular graph with these parameters. Another distance-
regular graph with the same parameters as the Hamming graph H(2m + n, 4) is the Doob graph
D(m, n), N =2m + n. MDS codes in Hamming graphs H(N,4) are classiﬁed in [5, 6].
AnMDScodeinthegraphD(m, n) with parameters ((m, n), 4
,d) is a code with code distance
d =2m + n − k + 1 and cardinality 4
. In the present paper we obtain a characterization of MDS
codes with code distance d ≥ 3 up to equivalence (characterization of the inﬁnite class of MDS
codes with distance 2 is separately considered in ). While in the graph H(N,4) for k =1there
exists a unique (up to equivalence) (n, 4
,d) MDS code, the number of such codes in the graph
D(m, n) is much greater. For 2m + n ≤ 6, the numbers of MDS codes with code distances d =3
and 4 are found, and all such codes code up to equivalence are listed in the Appendix. It is proved
that for 2m + n>6and2<d<2m + n no ((m, n), 4
,d) MDS codes exist. It is worth noting that
for MDS codes in the Hamming graph H(N, q) there is a well-known conjecture that MDS codes
with distance greater than 2 and less than N exist only for N ≤ q +2(for q =4,thisis6). For
linear codes this conjecture is proved in the case of a prime q; for nonprime values of q,important
result are obtained, which are close to a ﬁnal solution (see [8, 9]). As is seen from the results of the
present paper, a similar bound (2m + n ≤ 6 for MDS codes with distance greater than 2 but less
than 2m + n) is valid as well for MDS codes in Doob graphs.
Supported in part by the Russian Foundation for Basic Research, project no. 14-11-00555.