Optimal Equi-difference Conflict-avoiding Codes

Abstract

An equi-differece conflict-avoiding code $(CA{C}^e)\mathcal{C}$ of length $n$ and weight $\omega$ is a collection of $\omega$-subsets (called codewords) which has the form $\{0,i,2i,\cdots ,(\omega -1)i\}$ of ${\mathbb{Z}}_n$ such that $\Delta (c_1)\cap \Delta (c_2)=\varnothing$ holds for any $c_1,c_2\in \mathcal{C}$, $c_1\ne c_2$ where $Δ(c)=\{j-i \ (\mbox{mod}\ n) \; | \; i,j\in c,i\neq j\}.$ A code $\mathcal{C}\in CA{C}^{e}s$ with maximum code size for given $n$ and $\omega$ is called optimal and is said to be perfect if ${\cup }_{c\in \mathcal{C}}\Delta (c)={\mathbb{Z}}_n\backslash \{0\}.$ In this paper, we show how to combine a ${\mathcal{C}}_1\in CA{C}^e(q_1,\omega )$ and a ${\mathcal{C}}_2\in CA{C}^e(q_2,\omega )$ into a $\mathcal{C}\in CA{C}^e(q_1{q}_2,\omega )$ under certain conditions. One necessary condition for a $CA{C}^e$ of length $q_1{q}_2$ and weight $\omega$ being optimal is given. We also consider explicit construction of perfect $\mathcal{C}\in CA{C}^e(p,\omega )$ of odd prime $p$ and weight $\omega \ge 3$. Finally, for positive integer $k$ and prime $p\equiv1 \ (\mbox{mod}\ 4k)$, we consider explicit construction of quasi-perfect $\mathcal{C}\in CA{C}^e(2p,4k+1)$.

12 pages