Existence of 3 anti-cocircular truncated Möbius planes and constructions of strength-4 covering arrays
Kianoosh Shokri, Lucia Moura, Brett Stevens
TL;DR
The work generalizes the orthogoval framework to three anti-cocircular truncated Möbius planes on a common point set for odd $q$ and uses this geometry to construct strength-4 covering arrays with parameters $CA(3q^4-2;4,\frac{q^2+1}{2},q)$, improving known bounds for $q\ge11$. It achieves this via vertical concatenation of three strength-3 CAs associated with the truncated Möbius planes, rooted in ovoid-derived Möbius planes from $PG(3,q)$. A recursive construction then yields larger CAs, $CA(5q^4-4q^3-q^2+2q;4, q^2+1, q)$, broadening the practical impact for combinatorial testing. Throughout, the paper leverages finite-geometry tools (ovoids, Möbius planes, difference sets) to obtain explicit algebraic guarantees of coverage, demonstrating a strong link between geometric incidence structures and covering-array optimization.
Abstract
Two projective (affine) planes with the same point sets are orthogoval if the common intersection of any two lines, one from each, has size at most two. The existence of a pair of orthogoval projective planes has been proven and published independently many times. A strength-$t$ covering array, denoted by CA$(N; t, k, v)$, is an $N \times k$ array over a $v$-set such that in any $t$-set of columns, each $t$-tuple occurs at least once in a row. A pair of orthogoval projective planes can be used to construct a strength-$3$ covering array CA$(2q^3-1; 3, q^2 + q + 1, q)$. Our work extends this result to construct arrays of strength $4$. A $k$-cap in a projective geometry is a set of $k$ points no three of which are collinear. In $PG(3,q)$, an ovoid is a maximum-sized $k$-cap with $k =q^2+1$. Its plane sections (circles) are the blocks of a $3-(q^2 + 1, q + 1, 1)$ design, called a Möbius plane of order $q$. For $q$ an odd prime power, we prove the existence of three truncated Möbius planes, such that for any choice of these circles, one from each plane, their intersection size is at most three. From this, we construct a strength-$4$ covering array CA$(3q^4-2; 4, \frac{q^2+1}{2}, q)$. For $q \geq 11$, these covering arrays improve the size of the best-known covering arrays with the same parameters by almost 25 percent. The CA$(3q^4 -3; 4, \frac{q^2 +1}{2}, q)$ is used as the main ingredient in a recursive construction to obtain a CA$(5q^4 - 4q^3 - q^2 + 2q; 4, q^2 +1, q)$. Some improvements are obtained in the size of the best-known arrays using these covering arrays.