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Optimal binary covering codes of length 2j

✍ Scribed by William D. Weakley

Publisher
John Wiley and Sons
Year
2005
Tongue
English
Weight
144 KB
Volume
14
Category
Article
ISSN
1063-8539

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✦ Synopsis

Abstract

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j^, it is known that K(n,1) = 2 · K(nβˆ’1,1) = 2^nβ€‰βˆ’β€‰j^. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j^, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length nβ€‰βˆ’β€‰1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. Β© 2005 Wiley Periodicals, Inc. J Combin Designs

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