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All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [II, k] code. There is also a recent result on the complexity of computing the covering radius.doi:10.1109/tit.1985.1057043 fatcat:bq2zv7acnbapta45kd7lkbca7u