An Upper Bound of Resolution in Symmetrical Fractional Factorial Designs

Abstract

For the minimum weight in $s$-ary $(n, p)$ linear codes (or for the maximum resolution in $s^{n-p}$ designs), an upper bound has been obtained by Plotkin [4] where $s$ is a prime power. The purpose of this note is to obtain an improvement of the Plotkin's upper bound. Main result is as follows: when $p \geqq 2$, the maximum resolution $R_p(n, s)$ of any $s^{n-p}$ design satisfies the following: \begin{equation*}\begin{split}R_p(n, s) = s^{p-1}q\quad\text{if} m = 0, 1 \\ &\leqq s^{p-1}q + \lbrack s^{p-2}(s - 1)(m - 1)/(s^{p-1} - 1)\rbrack \\ \text{if} m = 2, 3, \cdots, s^{p-1} \\ &\leqq s^{p-1}q + \lbrack(s - 1)m/s\rbrack\quad\text{if} m = s^{p-1} + 1, \cdots, N - 1,\end{split}\end{equation*} where $\lbrack x\rbrack$ is the greatest integer not exceeding $x, n = qN + m$ and $N = (s^p - 1)/(s - 1)$.