Constructions and optimality results are given for block designs under first and second order (NN1 and NN2, respectively) neighbor correlations, extending the work of Kiefer and Wynn. Conditions for optimality and minimality are given for the NN2 model and new minimality results are found for the NN1 case. Construction of NN2 optimum complete block designs is solved and combinatorial arrays are used for NN2 optimum incomplete block designs. In many cases these are minimum optimum NN1 designs as well. A new solution for block size 3 is given. A method for constructing NN1 designs with partial variance balance is introduced and several series of these designs are shown to enjoy weaker optimality properties.
"Block Designs for First and Second Order Neighbor Correlations." Ann. Statist. 16 (3) 1206 - 1224, September, 1988. https://doi.org/10.1214/aos/1176350956