On Codes in Distance-Regular Graphs of Diameter 3

Let be a distance-regular graph of diameter d. For i ∈ {1, 2, . . . , d} the graph i is defined on the vertex set of and two vertices u, w are adjacent in i if and only if d(u,w) = i. The Shilla graph is a distance-regular graph of diameter 3 with the eigenvalue 1 = a3. For the Shilla graph the number a = a3 divides k and we set b = b() = k/a. The Shilla graph has intersection array {ab, (a+1)(b−1), b2; c1, c2, a(b−1)}. Jurisic and Vidali found intersection arrays of distance-regular graphs of diameter 3 containing the maximal locally regular 1-code perfect with respect to the last neighborhood. Moreover, such graph has intersection arrays {a(p + 1), cp, a + 1; 1, c, ap} (and is a strongly regular graph 3) or {a(p + 1), (a + 1)p, c; 1, c, ap} (and is a Shilla graph). In this manuscript we study graphs such that it contains the maximal locally regular 1-code. For a distance-regular graph with intersection array {a2, a2 −1, c; 1, c, a(a−1)} and a < 1000, c < 1000, the multiplicities of the eigenvalues are integer only in the cases (a, c) = (3, 4) (and q1 13 < 0), (a, c) = (5, 3), (a, c) = (9, 18) (and q3 33 < 0), (a, c) = (21, 49) (and q3 33 < 0), (a, c) = (21, 9). Thus, only arrays {25, 24, 3; 1, 3, 20} and {(441, 440, 9; 1, 9, 420)} remain. Moreover, a distance-regular graph with intersection array {a2, a2 − 1, c; 1, c, a(a − 1)} does not exist. As a consequence, distance-regular graphs with intersection arrays {25, 24, 3; 1, 3, 20} and {(441, 440, 9; 1, 9, 420)} do not exist.