Equidistant $q$-ary Codes with Maximal Distance and Resolvable Balanced Incomplete Block Designs

Equidistant $q$-ary codes with the maximal possible distance $d$ (for the given base $q$, number of words $N$, and number of digits $n$), called $ED_m$-codes, are considered. These $ED_m$-codes have parameters $N=qt$, $n=c(qt-1)/(q-1,t-1)$, $d=ct(q-1)/(q-1,t-1)$, where $c$ is an integer. The equivalence of $q$-ary $ED_m$-codes and resolvable balanced incomplete block designs is demonstrated. It is shown that extremal $ED_m$-codes with $n=(N-1)/(t-1)$ are equivalent to resolvable block designs with $\lambda=1$, and $ED_m$-codes with $n=(N-1)/(q-1)$ are equivalent to affine resolvable block designs and to complete orthogonal arrays of strength two.