Polygonal designs are introduced as a generalization of balanced incomplete block designs and as a specialization of partially balanced incomplete block designs. As in the case of balanced incomplete block designs, there is no hope of deciding the values of the parameters for which polygonal designs exist. We develop enough theory to reveal the structure, and thus, to resolve the existence problem for small polygonal designs, and derive necessary conditions for general cases.
 

AMS subject classification: 05B05, 62K10
 

Keywords: Balanced incomplete block designs; balanced sampling plans excluding contiguous units; Fisher's inequality.
 

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