Coding the vertexes of a graph

Given a graphGofnnodes. We wish to assign to each nodei(i = 1, 2, cdots n)a unique binary codec_{i}of lengthmsuch that, if we denote the Hannuing distance betweenc_{i}andc_{j}asH(c_{i}, c_{j}), thenH(c_{i}, c_{j})leq Tif nodesiandjare adjacent (i.e., connected by a single branch), andH(c_{i}, c_{j}) geq T+1otherwise. If such a code exists, then we say thatGis doable for the value ofTand tn associated with this code. In this paper we prove various properties relevent to these codes. In particular we prove 1) that for every graphGthere exists anmandTsuch thatGis doable, 2) for every value ofTthere exists a graphG?which is notTdoable, 3) ifGisT'doable, then it isT'+ 2pdoable forp = 0, 1, 2, cdots, and is doable for allT geq 2T'ifT'is odd, and is doable for allT geq 2T' + 1ifT'is even. In theory, the code can be synthesized by employing integer linear programming where eitherTand/ormcan be minimized; however, this procedure is computationally infeasible for values ofnandmin the range of about10or greater.