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Spectrum of sizes for perfect 2-deletion-correcting codes of length 4

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Abstract

Perfect \(t\)-deletion-correcting codes of length \(n\) over the alphabet of size \(q\), denoted by perfect \((n,t)_q\text {-DCCs}\), can have different number of codewords, because the balls of radius \(t\) with respect to Levenshteĭn distance may be of different sizes. Thus determining all possible sizes of a perfect \(t\)-deletion-correcting code makes sense. When \(t=n-2\), \(t\)-deletion-correcting codes are closely related to directed packings, constructions of which are based on the tools of design theory. Recently, Chee, Ge and Ling determined completely the spectrum of possible sizes for perfect \(q\)-ary 1-deletion-correcting codes of length three for all \(q\), and perfect \(q\)-ary 2-deletion-correcting codes of length four for all but \(19\) values of \(q\). In this paper, we continue to investigate the spectrum problem for perfect \((4,2)_q\text {-DCCs}\). By constructing a considerable number of incomplete directed packings, we give an almost complete solution to the spectrum problem of sizes for perfect \((4,2)_q\text {-DCCs}\), leaving the existence of \((4,2)_{19}\text {-DCC}\) of size \(62\) and \((4,2)_{34}\text {-DCC}\) of size \(196\) in doubt.

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Acknowledgments

This research supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001. The authors express their gratitude to the two anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the technical presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Hengjia Wei

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Gennian Ge

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  1. Hengjia Wei
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  2. Gennian Ge
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Correspondence to Gennian Ge.

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Communicated by D. Jungnickel.

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Wei, H., Ge, G. Spectrum of sizes for perfect 2-deletion-correcting codes of length 4. Des. Codes Cryptogr. 74, 127–151 (2015). https://doi.org/10.1007/s10623-013-9848-x

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  • DOI: https://doi.org/10.1007/s10623-013-9848-x

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