Abstract
A conflict-avoiding code (CAC) of length \(n\) and weight \(w\) is defined as a family \({\mathcal C}\) of \(w\)-subsets (called codewords) of \({\mathbb {Z}}_n\), the ring of residues modulo \(n\), such that \(\Delta (C) \cap \Delta (C') = \emptyset \) for any \(C, C' \in {\mathcal C}\), where \(\Delta (C) = \{ j-i \pmod {n} : i, j \in C, i \ne j\}\). A code \({\mathcal C}\) in CACs of length \(n\) and weight \(w\) is called an equi-difference code if every codeword \(C \in {\mathcal C}\) has the form \(\{ 0, i, 2i, \ldots , (w-1) i \}\). A code \({\mathcal C}\) in CACs of length \(n\) and weight \(w\) is said to be optimal if \({\mathcal C}\) has the maximum number of codewords. In this article, we investigate sizes and constructions of optimal codes in equi-difference CACs of weight four by using properly defined directed graphs. As a consequence, several series of infinite number of optimal equi-difference CACs are also provided.
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Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and helpful advice. The work of M. Mishima was supported in part by JSPS under Grant-in-Aid for Scientific Research (C)25400200 and the work of M. Jimbo was supported in part by JSPS under Grant-in-Aid for Scientific Research (B)22340016 and Grant-in-Aid for Challenging Exploratory Research 26610036.
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Communicated by C. Mitchell.
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Lin, Y., Mishima, M. & Jimbo, M. Optimal equi-difference conflict-avoiding codes of weight four. Des. Codes Cryptogr. 78, 747–776 (2016). https://doi.org/10.1007/s10623-014-0030-x
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DOI: https://doi.org/10.1007/s10623-014-0030-x