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In this paper new infinite families of linear binary completely transitivecodes are presented. They have covering radius $\rho = 3$ and 4, and are a halfpart of the binary Hamming and the binary extended Hamming code of length$n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completelytransitive codes, in the usual way, i.e., as coset graphs, new presentations ofinfinite families of distance transitive coset graphs of diameter three andfour, respectively, are constructed.

New families of completely transitive codes and distance transitive graphs