A Detection Matrix for $3N^n$ Search Design

A parallel flats fraction for the $3^n$ factorial experiment is defined as the union of flats, ${t$\mid$At=C_i(mod 3)}, i=1,2,\cdot,f$, in EG(n,3) and is symbolically written as At=C where A is of rank r. The A matrix partitions the effects into u+1 alias sets where $u=(3^{n-r}-1)/2$. For each alias set the f flats produce an alias component permutation matrix (ACPM) with elements from $S_3$. In this paper, a detection vector of the ACPM was constructed for each combination of k or fewer two-factor interactions. Also the relationship between the detection vectors has been shown.