Codes in distance-regular graphs with theta(2) = -1

If a distance-regular graph Gamma of diameter 3 contains a maximal 1-code C that is both locally regular and last subconstituent perfect, then 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 = p(33)(3) (Jurisic and Vidali). In 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 graphs with intersection array {a(p+1), cp, a+1; 1, c, ap} in which any two vertices at distance 3 are in a maximal 1-code. In particular, we find four new infinite families of intersection arrays: {a(a - 2), (a - 1)(a - 3), a + 1; 1, a - 1, a(a - 3)} for a >= 5, {a(2a + 3), 2(a - 1)(a + 1), a + 1; 1, a - 1, 2a(a + 1)} for a not congruent to 1 modulo 3, {a(2a - 3), 2(a - 1)(a - 2), a+1; 1, a - 1, 2a(a - 2)} for even a not congruent to 1 modulo 3, and {a(3a - 4), (a - 1)(3a - 5), a + 1; 1, a - 1, a(3a - 5)} for even a congruent to 0 or 2 modulo 5.