Codes in Shilla Distance-Regular Graphs

Let Gamma be a distance-regular graph of diameter 3 containing a maximal 1-code C, which is locally regular and last subconstituent perfect. Then the graph Gamma has intersection array {a(p + 1), cp, a + 1; 1,c, ap} or {a(p + 1), (a + 1)p, c;1, c, ap}, where a = a(3), c = c(2), and p = p333 (Jurisic, Vidali). In the first case, Gamma has eigenvalue theta(2) = -1 and the graph Gamma(3) is pseudogeometric for GQ(p + 1, a). In the second case, Gamma is a Shilla graph. We study Shilla graphs in which every two vertices at distance 2 belong to a maximal 1-code. It is proved that, in the case theta(2) = -1, a graph with the specified property is either the Hamming graph H(3, 3) or a Johnson graph. We find necessary conditions for the existence of Q-polynomial Shilla graphs in which any two vertices at distance 3 lie in a maximal 1-code. In particular, we find two infinite families of feasible intersection arrays of Q-polynomial graphs with the specified property: {b(b(2) - 3b)/2, (b - 2)(b - 1)(2)/2, (b - 2)t/2; 1, bt/2, (b(2) - 3b)(b - 1)/2} (graphs with p333 = 0) and {b(2)(b - 4)/2, (b(2) - 4b + 2)(b - 1)/2, (b - 2)l/2; 1, bl/2, (b(2) - 4b)(b - 1)/2} (graphs with p333 = 1).