An equi-differece conflict-avoiding code $(CAC^{ e})\ \mathcal{ C}$ 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 $\Delta(c_{ 1})\cap\Delta(c_{ 2})=\emptyset$ holds for any $c_{ 1},\ c_{ 2}\in\mathcal{ C}$, $c_{ 1}\neq c_{ 2}$ where $\Delta(c)=\{ j-i \ (\mbox{ mod}\ n) \; | \; i,j\in c,i\neq j\}.$ A code $\mathcal{ C}\in CAC^{ e}s$ 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 $\mathcal{ C}_{ 1}\in CAC^{ e}(q_{ 1},\omega)$ and a $\mathcal{ C}_{ 2}\in CAC^{ e}(q_{ 2},\omega)$ into a $\mathcal{ C}\in CAC^{ e}(q_{ 1}q_{ 2},\omega)$ under certain conditions. One necessary condition for a $CAC^{ e}$ of length $q_{ 1}q_{ 2}$ and weight $\omega$ being optimal is given. We also consider explicit construction of perfect $\mathcal{ C}\in CAC^{ e}(p,\omega)$ of odd prime $p$ and weight $\omega\geq3$. Finally, for positive integer $k$ and prime $p\equiv1 \ (\mbox{ mod}\ 4k)$, we consider explicit construction of quasi-perfect $\mathcal{ C}\in CAC^{ e}(2p,4k+1)$.
Categories: cs.IT math.IT