For a given number of varieties v, number of blocks b, common treatment replication r and common block size k, rv = bk, 3 <= k <= 8, k <= v, necessary conditions for the existence of p-trend-free equireplicated binary block design are derived, where 2 <= p <= k-1. The conditions are also sufficient for complete block designs (where v=k and b=r). For balanced incomplete block designs (BIBDs) that satisfy the necessary conditions, we investigate whether varieties can be ordered within blocks to make the design p-trend- free. For v <= 20 and b <= 100, if the necessary conditions are satisfied and if a BIBD exists, we conclude that a p-trend-free BIBD can also be found.
 

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