On the covering radius of codes

The covering radius R of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning t[n, k] , the minimal covering radius of any binary code of length n and dimension k . For example t[n, 4] and t[n, 5] are determined exactly, and reasonably tight bounds on t[n, k] are obtained for any k when n is large. These results are found by using several new constructions for codes with small covering radius. One construction, the amalgamated direct sum, involves a quantity called the norm of a code. Codes with norm \leq 2 R + 1 are called normal, and may be combined efficiently. The paper concludes with a table giving bounds on t[n, k] for n \leq 64 .

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