Trend-free block designs for higher order trends

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 designs are derived for 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 designs 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 is also available.