OBIBD construction for designs with λ = 1 and odd k = 5.
Let v be a prime number or power of the form 20t+1, and x be a primitive root of the Galois Field of order v.
  • If (x4t - 1), (x8t - 1), (x12t - 1), (x16t - 1) are, say, xp, xq, xr, xs, and p,q,r,s are all distinct, modulo 4, then (x4i, x4t+4i,x8t+ 4i, x12t+4i, x16t+4i) for i = 0.. (t-1), define a set of initial blocks for a BIBD with the required parameters. (This is not Bose's construction.)
  • If there exists an integer m such that (xm-1),(x4t+m-1), (x8t+m-1), (x12t+m-1), and (x16t+m-1) are, say, xp, xq, xr, xs and xt,and p,q,r,s and t all equal, modulo 4, then an OBIBD(v) can be constructed from the initial blocks: (x4i, x4i+m; x4t+4i, x4t+4i+m; x8t+4i, x8t+4i+m; x12t+4i, x12t+4i+m; x16t+4i, x16t+4i+m), for i = 0,1,...,(t-1)
(The full design is obtained by developing over the elements of the Galois Field.)

Proof.Thedifference squareformed by the typical block is exactly anologous to that formed for the case with k = 3. The rest of the proof therefore follows.

The design relating the two treatment sets is not, in general, a SBIBD, so the design just constructed is an OBIBD not a pergola.

Several values of m have been found for designs with k = 5: tables here show the values found.

A similar construction for designs with v = 42t+1, k = 7, λ = 1 allows values of m as shown here. For these designs, the first initial block is multiplied by x6, x12 etc rather than x4, x8, etc to give subsequent initial blocks.

There are SBIBDs based on biquadratic residues: these require v to be of the form 4t2+1, where t is an odd positive integer, and v is prime (HCD I.5.6). This means v is of the form 16s(s+1)+5, which seems to be of the form 20t+1 when s is of the form 5p+2. So this gives a formula of 100(2p+1)2 + 1, thus 101, 901, 2501, 4901, etc. None of the values examined gave a BIBD, however.