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Erschienen in:
01.03.2016
Optimal equi-difference conflict-avoiding codes of weight four
Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2016
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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.