Abstract
A conflict-avoiding code (CAC) \(\mathcal {C}\) of length \(n\) and weight \(k\) is a collection of \(k\)-subsets of \(\mathbb {Z}_n\) such that \(\varDelta (x)\cap \varDelta (y)=\emptyset \) for any \(x,y\in \mathcal {C}\) and \(x\ne y\), where \(\varDelta (x)=\{a-b:\, a,b\in x, a\ne b\}\). Let \(\text {CAC}(n,k)\) denote the class of all CACs of length \(n\) and weight \(k\). A CAC \(\mathcal {C}\in \text {CAC}(n,k)\) is said to be equi-difference if any codeword \(x\in \mathcal {C}\) has the form \(\{ 0,i,2i,\ldots , (k-1)i \}\). A CAC with maximum size is called optimal. In this paper we propose a graphical characterization of an equi-difference CAC, and then provide an infinite number of optimal equi-difference CACs for weight four.
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Fu H.L., Lin Y.H., Mishima M.: Optimal conflict-avoiding codes of even length and weight 3. IEEE Trans. Inf. Theory 56(11), 5747–5756 (2010).
Fu H.L., Lo Y.H., Shum K.W.: Optimal conflict-avoiding codes of odd length and weight three. Des. Codes Cryptogr. 1–21 (2012). doi:10.1007/s10623-012-9764-5.
Györfi L., Vajda I.: Constructions of protocol sequences for multiple access collision channel without feedback. IEEE Trans. Inf. Theory 39(5), 1762–1765 (1993).
Jimbo M., Mishima M., Janiszewski S., Teymorian A.Y., Tonchev V.: On conflict-avoiding codes of length \(n=4m\) for three active users. IEEE Trans. Inf. Theory 53(8), 2732–2742 (2007).
Levenshtein V.I.: Conflict-avoiding codes and cyclic triple systems. Probl. Inf. Transm. 43(3), 199–212 (2007).
Levenshtein V.I., Tonchev V.D.: Optimal conflict-avoiding codes for three active users. In: Proceedings of IEEE Internation Symposium on Information Theory, Adelaide, Australia. pp. 535–537. (2005).
Lin Y., Mishima M., Satoh J., Jimbo M.: Optimal equi-difference conflict-avoiding codes of odd length and weight three. Finite Fields Appl. 26, 49–68 (2014).
Ma W., Zhao C., Shen D.: New optimal constructions of conflict-avoiding codes of odd length and weight 3. Des. Codes Cryptogr. 1–14 (2013). doi:10.1007/s10623-013-9827-2.
Mishima M., Fu H.L., Uruno S.: Optimal conflict-avoiding codes of length \(n\equiv 0\) (mod 16) and weight 3. Des. Codes Cryptogr. 52(3), 275–291 (2009).
Momihara K.: Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three. Des. Codes Cryptogr. 45(3), 379–390 (2007).
Momihara K., Müller M., Satoh J., Jimbo M.: Constant weight conflict-avoiding codes. SIAM J. Discret. Math. 21(4), 959–979 (2007).
The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences. http://oeis.org.
Shum K.W., Wong W.S., Chen C.S.: A general upper bound on the size of constant-weight conflict-avoiding codes. IEEE Trans. Inf. Theory 56(7), 3265–3276 (2010).
Shum K.W., Wong W.S.: A tight asymptotic bound on the size of constant-weight conflict-avoiding codes. Des. Codes Cryptogr. 57(1), 1–14 (2010).
Wu S.L., Fu H.L.: Optimal tight equi-difference conflict-avoiding codes of length \(n=2^k\pm 1\) and weight 3. J. Comb. Des. 21(6), 223–231 (2013).
Acknowledgments
The authors would like to express their gratitude to the referees for their valuable comments and suggestions in improving the presentation of this paper. Partially supported by National Science Council, Taiwan under grants NSC 101-2115-M-390-004-MY3 (Y. H. Lo) and NSC 100-2115-M-009-005-MY3 (H. L. Fu and Y. H. Lin).
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Communicated by V. D. Tonchev.
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Lo, YH., Fu, HL. & Lin, YH. Weighted maximum matchings and optimal equi-difference conflict-avoiding codes. Des. Codes Cryptogr. 76, 361–372 (2015). https://doi.org/10.1007/s10623-014-9961-5
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DOI: https://doi.org/10.1007/s10623-014-9961-5