Optimal Tight Equi‐Difference Conflict‐Avoiding Codes of Length n = 2k ± 1 and Weight 3

For a k-subset X of , the set of differences on X is the set (mod n): . A conflict-avoiding code CAC of length n and weight k is a collection of k-subsets of such that = ∅ for any distinct . Let CAC() be the class of all the CACs of length n and weight k. The maximum size of codes in CAC(n, k) is denoted by . A code CAC(n, k) is said to be optimal if = . An optimal code is tight equi-difference if = and each codeword in is of the form . In this paper, the necessary and sufficient conditions for the existence problem of optimal tight equi-difference conflict-avoiding codes of length n = and weight 3 are given. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 223–231, 2013

[1]  Vladimir D. Tonchev,et al.  On Conflict-Avoiding Codes of Length $n=4m$ for Three Active Users , 2007, IEEE Transactions on Information Theory.

[2]  László Györfi,et al.  Constructions of protocol sequences for multiple access collision channel without feedback , 1993, IEEE Trans. Inf. Theory.

[3]  Peter Mathys,et al.  A class of codes for a T active users out of N multiple-access communication system , 1990, IEEE Trans. Inf. Theory.

[4]  László Györfi,et al.  Constructions of binary constant-weight cyclic codes and cyclically permutable codes , 1992, IEEE Trans. Inf. Theory.

[5]  Kenneth W. Shum,et al.  Optimal conflict-avoiding codes of odd length and weight three , 2014, Des. Codes Cryptogr..

[6]  Vladimir I. Levenshtein,et al.  Conflict-avoiding codes and cyclic triple systems , 2007, Probl. Inf. Transm..

[7]  Kenneth W. Shum,et al.  A tight asymptotic bound on the size of constant-weight conflict-avoiding codes , 2010, Des. Codes Cryptogr..

[8]  Koji Momihara,et al.  Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three , 2007, Des. Codes Cryptogr..

[9]  Hung-Lin Fu,et al.  Optimal conflict-avoiding codes of length n ≡ 0 (mod 16) and weight 3 , 2009, Des. Codes Cryptogr..

[10]  Vladimir D. Tonchev,et al.  Optimal conflict-avoiding codes for three active users , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[11]  Hung-Lin Fu,et al.  Optimal Conflict-Avoiding Codes of Even Length and Weight 3 , 2010, IEEE Transactions on Information Theory.

[12]  Meinard Müller,et al.  Constant Weight Conflict-Avoiding Codes , 2007, SIAM J. Discret. Math..

[13]  Hung-Lin Fu,et al.  Optimal Conflict-Avoiding Codes of Even Length and Weight 3 , 2010, IEEE Transactions on Information Theory.

[14]  Kenneth W. Shum,et al.  A General Upper Bound on the Size of Constant-Weight Conflict-Avoiding Codes , 2010, IEEE Transactions on Information Theory.

[15]  Koji Momihara.  On cyclic 2(k-1)-support (n, k)k-1 difference families , 2009, Finite Fields Their Appl..

[16]  Boris Tsybakov,et al.  Some Constructions of Conflict-Avoiding Codes , 2002, Probl. Inf. Transm..