Permuting a codeword generates lists of strings that constitute bases of a permutation group. This paper is a portion (with amplification) of Bailey’s 2005 PhD mathematics dissertation, “Permutation Groups, Error Correcting Codes, and Uncoverings,” in which he describes the use of permutation groups for error correcting codes. The paper focuses on the theory of uncovering-by-bases and constructions based on the theory. An uncovering-by-bases is a set of bases such that any combination of error positions is avoided. The constructions are generated for all rank-2 base-transitive groups, plus additional constructions for some groups of higher rank. (A rank-1 base-transitive group is a regular permutation group.)

As a presentation of the theoretical foundations underlying uncovering-by-bases, this paper appears to be complete and well organized. However, since it is a portion of a dissertation, it lacks the context of applicability to computer error correction. The latter is found in a companion paper (still available in preprint form as of this writing), and in the complete dissertation. Fortunately, both are available for download from the author’s Web site, given in the paper. The place to begin is the dissertation. Those most likely to benefit from this paper are those with a sophisticated background in group theory and combinatorics, basically, research mathematicians.