Codes in Bipartite Distance^regular Graphs

For each bipartition_of a bipartite distance-regular graph F, there naturally corresponds another distance-regular graph F called a halved graph. It is shown that the existence of a perfect e-code in a halved graph F is equivalent to the existence of a uniformly packed 2e-code in F with certain specific parameters. Using this equivalence, we show the non-existence of perfect codes for two classes of distance-regular graphs F corresponding to F = Qk and F = 2. Ok. For the basic definitions and properties of distance-regular graphs, the reader is referred to Biggs [2]. Let T be a distance-regular graph with distance function d and intersection array '" 1 r r r\ 1 L2 ... < d i C 1). We say that C has external distance (i.e. the true external distance in the sense of Delsarte [3]) e + m if the maximal distance of any vertex of F from C is e+m. We choose zi (e V(T), the set of vertices of F) such that d(zj, C) = j (where j e {0, 1, ..., e+m}) and call C completely regular if the numbers |{xeC| d(x,zj) = i}\ = Pij(C,zj) = py(C) (where ie{0, 1, ...}d} and j e {0, 1, ..., e + m}) depend only on i and j and not on the choice of Zj. We say that C is locally regular if the numbers py(C,zj) = Pij(C) (where i , je{0, 1, ...}e+m}) depend only on i and ; and not on the choice of zy It is proved that a locally regular Received 5 November, 1976. [J. LONDON MATH. SOC. (2), 16 (1977), 197-202]