OBIBD construction for Netto designs

Let v be a prime number or power of the form 12t+7, and x be a primitive root of the Galois Field of order v. If there exists an integer m such that (x^m-1),(x^4t+m+2-1) and (x^8t+m+4-1) are all odd, or all even, powers of x, then an OBIBD(v) can be constructed from the initial blocks:

(x²ⁱ,x^2i+m; x^4t+2i+2,x^{4t+2i+ m+2};x^8t+2i+4,x ^8t+2i+m+4), for i = 0,1,...,2t

(The full design is obtained by developing the initial blocks over the elements of the Galois Field.)

Proof.The difference square formed by the typical block is:

where x^a=(x^m-1), x^b=(x^4t+m+2-1), and x^c=(x^8t+m+4-1).

If all of these are odd(even), then all the differences in the typical square will be odd(even), and the complete set of all differences generated will comprise 3 occurrences of each of the odd(even) powers of x. The elements on the leading diagonals will comprise one occurrence of each (being multiples of a fixed element by all the even powers of the primitive root).

The design generated for the second set of treatments is a BIBD. The differences generated in the initial blocks will be exactly the same as those for the first set of treatments, save that they are multiplied by x^m throughout.