Moore-Greig Designs I

This is the first in a series of three papers in which we investigate a special class of designs that we designate as “Moore-Greig Designs”. The sobriquet is associated with the fact that ideas gleaned from two constructions, one due to E. H. Moore (1896) and the other due to M. Greig (2003), are combined to produce designs that have remarkable properties and features. A Moore-Greig Design is an RBIBD that contains, simultaneously, nested RBIBDs, nested GWhDs, many GWhDs, frames, nested frames, GWhFrames, nested GWhFrames, GWhaFrames, RRDFs, and nested RRDFs. All of these designs are Z-cyclic.

To be more precise, let \( p \) be a prime and let \( \{s_i\}_{i=1}^m \) be a monotone increasing sequence of positive integers such that \( s_i | s_{i+1} \) for all \( i, 1 \leq i \leq m-1 \). Let \( n \) be a positive integer such that \( s_m \leq n \) and \( s_m | n \). A Moore-Greig Design is a Z-cyclic \( (p^n, p^{s_m}, p^{s_m} – 1) \)-RBIBD that contains: (1) a Z-cyclic \( (p^n, p^{s_i}, p^{s_i} – 1) \)-RBIBD for each \( i, 1 \leq i \leq m-1 \), (2) a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhD\( (p^n) \) for each \( i, 1 \leq i \leq m-1 \), (3) for each \( i, 1 \leq i \leq m-1 \), a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhD\(_a(p^n)\) for each \( a = \frac{\alpha}{p^{s_i} – 1} \), \( \alpha = 1, 2, \ldots, p^{s_i} – 2 \), (4) a Z-cyclic \( \{p^{s_m}\} \)-frame of type \( (p^{s_m} – 1)^q \), where \( q = \frac{p^{n} – 1}{p^{s_m} – 1} \), (5) a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhFrame of type \( (p^{s_m} – 1)^q \) for each \( i, 1 \leq i \leq m-1 \), (6) a Z-cyclic \( (p^{n} – 1, p^{s_i} – 1, p^{s_i}, 1) \)-RRDF for each \( i, 1 \leq i \leq m \).

Other than a single published example, there is no literature pertaining to GWhDs. Therefore, the infinite classes of GWhDs constructed from the Moore-Greig Designs are the first general results related to this type of design. It is also believed that many of the other designs contained within the infinite classes of Moore-Greig designs are new.

In this paper, Part I, we provide detailed descriptions of both the Moore construction and the Greig construction. In the case of the Moore construction, we supply proofs since such proofs are lacking in Moore’s paper. Also included in this paper is a description of Moore-Greig Designs corresponding to \( m = 2 \) and a discussion is given of the presence of the GWhFrames, nested designs, and the RRDFs. Our methods are such that the constructions are straightforward if one has the (associated) Galois Field.