Designs from projective planes and PBD bases

Abstract

The essential theme of this article is the exploitation of known configurations in projective planes in order to construct pairwise balanced designs. Among the results shown are (q 2 + 1)/2 ∈ B((q + 1)/2, 2) for all odd prime powers q; the resolution of 31 of the Mullin and Stinson's open cases in the spectrum of B(P7, 1), where P7 is odd prime powers ≥ 7; some constructions showing the inessential nature of values in E a + , where E a + are PBD generating sets containing values equal to 1 mod(a) for 5 ≤ a ≤ 7; some constructions with block sizes being prime powers ≥ 8, yielding 7 MOLS of order 158 and 9 MOLS of order 254; and some designs on 28, 38, 42, 56, 63, 68, 72, and 156 points with interesting block sizes. Remark 4.2. Useful examples where 1 < t < k − 1 are not common. Bennett [11] used this in the T [17, 1; 31] case, and filled in the groups with a point at infinity to show 528 − t ∈ B({16, 17, 17 − t * , 31, 32}, 1). Another example deletes three collinear points in a T [25, 1; 31] giving 772 ∈ B({22 * , 24, 25, 30, 31}, 1) as an intermediate step in constructing a B[6, 1; 5 · 772 + 1], a value that was not constructed in [50], but has since been constructed by other means [64].If we have the basic design embedded as an arc in a projective plane, then we also have the option of extending, or spiking, any line of the plane up to its full size in the plane. Theorem 4.3. If a projective plane of order q contains a {v; K ∪ k * }-arc, and 0Remark 4.4. Note that whether {1} ∈ K or not is not inconsequential if blocks of size 2 are to be avoided.Remark 4.5. This construction has often been used with transversal designs, and the most usual construction takes a point not on the special line to generate the groups, yieldingCorollary 4.6. 123 ∈ B({8, 9, 11 * }, 1).Proof. We may embed the 120 point Seiden design (discussed in Section 5) in PG(2, 16), and take t = 3 and k = 8.Remark 4.7. This value was unknown in [11].Usually the two constructions in Theorems 4.1 and 4.3 are not mixed, as no new v values result, and extra block sizes are generated. However, when we can stand the extra block sizes, we can sometimes achieve a specific block size with a specific v. This is useful in constructing incomplete transversal designs. Theorem 4.8. If v ∈ GD(K, λ, M ) and K ⊂ RT (s, 1) and M ⊂ T (s, λ), then v ∈ T (s, λ).Proof. See [38, Theorem 3.2]. The proof also shows that if a group of size m occurs, then the design will contain T [k, λ; m] as a subdesign.Remark 4.9. Wilson [62] gives an alternative construction which is typically better than Theorem 4.8. However, Theorem 4.8 suffices here; I was not able to show that I could satisfy Wilson's hypotheses for some of the cases I needed. Lemma 4.10. If 7 ≤ a ≤ 18, then 160 + a ∈ B({7, 8, 9, 23} ∪ 13 * ∪ a * , 1).Proof. Start with a T [7, 1, 23], embedded in PG(2, 23), and select two lines that intersect in this design, and apply Theorem 4.3 twice, once with t = 6, and once with t = a − 7. If the groups of the design were generated as in Remark 4.5, since 6 + a ≤ 24, we can en...

Pair covering and other designs with block size 6

“…The essential elements listed then follow by the integrality conditions for a (v, 6, 1) BIBD, from Fisher's rank condition, the Bruck-Chowla-Ryser condition, or from [26]. The remaining improvements on results from [12,24,30] In [12], Abel and Greig updated the earlier results of Mullin and of Greig on the PBD closure of Q = {6} ∪ {ṗrime powers ≡ 1 (mod 5)}. We improve this result.…”

“…As before, no two points in any given block of size 5 have identical first components (mod 5); thus each of these blocks generate 5 parallel classes on the non-infinite points and we can add one infinite point to each. 20 9 10 1 : ((0, 0), (0, 1), (0, 12), (0, 29), (1,16), (3,24)) , ((0, 0), (0, 3), (0, 16), (1,13), (3,10), (3,32)) , ((0, 0), (0, 2), (0, 32), (1,24), (1,34), (2,17)) , ((0, 0), (1, 1), (2, 31), (3, 2), (4, 5)) , ((0, 0), (0, 15), (1,21), (1,26), (2, 2), (3, 1)) , ((0, 0), (1,5), (2,25), (3,33), (4,11)) . 10 22 15 1 : ((0, 0), (0, 3), (0, 4), (0, 19), (0, 30), (1,5)) , ((0, 0), (0, 2), (0, 7), (0, 38), (2, 3), (2, 15)) , ((0, 0), (0, 10), (1,21), (1,41), (2,34), (4,29)) ,((0, 0), (1,3), (2,29), (3,9), (4, 5)) , ((0, 0), (0, 9),…”

“…Very little work on 2-(v, 6, 1) covers has been done: the small values (i.e., v ≤ 32) have been studied [16,33] and C(v, 6, 2) = L(v, 6, 2) in most of the known cases, but C(30, 6, 2) = L(30, 6, 2) + 1; C(v, 6, 2) = L(v, 6, 2) + 2 for v ∈ {11, 15, 16, 19, 20, 25, 26}; C(21, 6, 2) = L(21, 6, 2) + 3; L(23, 6, 2) ≤ C(23, 6, 2) ≤ L(23, 6, 2)+1; L(24, 6, 2)+1 ≤ C (24,6,2) ≤ L(24, 6, 2)+2; L(29, 6, 2) ≤ C (29,6,2) ≤ L(29, 6, 2) + 1.…”

Pairwise balanced designs whose block size set contains seven and thirteen

“…They showed that B(P 1,6 ) ∪ Q = N 1,6 , where Q is a set of 31 (possible) exceptions. Subsequently, their result was improved by Greig [17] who removed 9 of the values in Q. In Section 4, a number of new constructions, which are based on Wilsons Fundamental Construction, are introduced.…”

“…Mullin [21] determined a first Wilson basis for the set N 1,6 = B({7, 13, 19, 25, 31, 37, 43, 55, 61, 67, 73, 79, 97, 103, 109, 115, 121, 127, 139, 145, 157, 163, 181, 1 modulo 42 1 7 13 19 25 31 37 Largest exception 2605 1645 14293 82549 98683 91507 88447 193, 199, 205, 211, 223, 229, 235, 241, 253, 265, 271, 277, 283, 289, 295, 307, 313, 319, 331, 349, 355, 361, 367, 373, 379, 391, 397, 409, 415, 421, 439, 445, 451, 457, 487, 493, 499, 643, 649, 655, 661, 667, 685, 691, 697, 709, 727, 733, 739, 745, 751, 781, 787, 811, 1063, 1069, 1231, 1237, 1243, 1249, 1255, 1315, 1321, 1327, 1543, 1549, 1567, 1579 1585, 1783, 1789, 1795, 1801, 1819, 1831}). Independently, Du [14] and Greig [17] removed the elements 223, 253,295,307,361,367,379,421,439,655,727,1231,1237,1243,1249,1255,1543,1549,1585,1783,1789,1795,1801,1819, and 1831 from the finite basis for N 1,6 . We will improve this result in Section 4 by showing that 4 further values 1063, 1069, 1567, and 1579 are not essential.…”