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Classification of the odd sets in \(\hbox {PG}(4,4)\) and its application to coding theory

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Abstract

A set \({\fancyscript{K}}\) in \(\hbox {PG}(r,4), r \ge 2\), is odd if every line meets \({\fancyscript{K}}\) in an odd number of points. We show there are exactly 45 inequivalent odd sets in \(\hbox {PG}(4,4)\) up to projective equivalence. As an application to coding theory, a new sufficient condition for the extendability of quaternary linear codes is given.

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, Osaka, 599-8531, Japan

    Taichiro Tanaka & Tatsuya Maruta

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  1. Taichiro Tanaka
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  2. Tatsuya Maruta
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Correspondence to Tatsuya Maruta.

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The second author is partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 24540138.

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Tanaka, T., Maruta, T. Classification of the odd sets in \(\hbox {PG}(4,4)\) and its application to coding theory. AAECC 24, 179–196 (2013). https://doi.org/10.1007/s00200-013-0201-4

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  • DOI: https://doi.org/10.1007/s00200-013-0201-4

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Mathematics Subject Classification (2000)