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We describe how previously known methods for determining the number of decimation classes of density $\delta$ binary vectors can be extended to nonnegative integer vectors, where the vectors are indexed by a finite abelian group $G$ of size $\ell$ and exponent $\ell^*$ such that $\delta$ is relatively prime to $\ell^*$. We extend the previously discovered theory of multipliers for arbitrary subsets of finite abelian groups, to arbitrary multisets of finite abelian groups. Moreover, this developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing., Comment: 13 pages

We prove the equivalence of two-symbol supersaturated designs (SSDs) with $N$ (even) rows, $m$ columns, $s_{\rm max} = 4t +i$, where $i\in\{0,2\}$, $t \in \mathbb{Z}^{\geq 0}$ and resolvable incomplete block designs (RIBDs) whose any two blocks intersect in at most $(N+4t+i)/4$ points. Using this equivalence, we formulate the search for two-symbol E($s^2$)-optimal and minimax-optimal SSDs with $s_{\max} \in \{2,4,6\}$ as a search for RIBDs whose blocks intersect accordingly. This allows developing a bit-parallel tabu search (TS) algorithm. The TS algorithm found E($s^2$)-optimal and minimax-optimal SSDs achieving the sharpest known E($s^2$) lower bound with $s_{\max} \in \{2,4,6\}$ of sizes $(N,m)=(16,25), (16,26), (16,27), (18,23),(18,24),(18,25),(18,26),(18,27),(18, 28),$ $(18,29),(20,21),(22,22),(22,23),(24,24)$, and $(24,25)$. In each of these cases no such SSD could previously be found.

The Legendre pair problem is a particular case of a rank-$1$ semidefinite description problem that seeks to find a pair of vectors $(\mathbf{u},\mathbf{v})$ each of length $\ell$ such that the vector $(\mathbf{u}^{\top},\mathbf{v}^{\top})^{\top}$ satisfies the rank-$1$ semidefinite description. The group $(\mathbb{Z}_\ell\times\mathbb{Z}_\ell)\rtimes \mathbb{Z}^{\times}_\ell$ acts on the solutions satisfying the rank-$1$ semidefinite description by $((i,j),k)(\mathbf{u},\mathbf{v})=((i,k)\mathbf{u},(j,k)\mathbf{v})$ for each $((i,j),k) \in (\mathbb{Z}_\ell\times\mathbb{Z}_\ell)\rtimes \mathbb{Z}^{\times}_\ell$. By applying the methods based on representation theory in Bulutoglu [Discrete Optim. 45 (2022)], and results in Ingleton [Journal of the London Mathematical Society s(1-31) (1956), 445-460] and Lam and Leung [Journal of Algebra 224 (2000), 91-109], for a given solution $(\mathbf{u}^{\top},\mathbf{v}^{\top})^{\top}$ satisfying the rank-$1$ semidefinite description, we show that the dimension of the convex hull of the orbit of $\mathbf{u}$ under the action of $\mathbb{Z}_{\ell}$ or $\mathbb{Z}_\ell\rtimes\mathbb{Z}^{\times}_\ell$ is $\ell-1$ provided that $\ell=p^n$ or $\ell=pq^i$ for $i=1,2$, any positive integer $n$, and any two odd primes $p,q$. Our results lead to the conjecture that this dimension is $\ell-1$ in both cases. We also show that the dimension of the convex hull of all feasible points of the Legendre pair problem of length $\ell$ is $2\ell-2$ provided that it has at least one feasible point., Comment: 27 pages

By assuming a type of balance for length $\ell=87$ and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length $\ell=85$, we find LPs of these lengths. We then study the power spectral density (PSD) values of m-compressions of LPs of length 5m. We also formulate a conjecture for Legendre pairs of lengths $\ell \equiv 0$ (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range $\leq 200$ for which the existence question of LPs remains unsolved from 12 to 10.

There is always a natural embedding of $S_s\wr S_k$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$ level, strength $1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_2\wr S_k$ for all $k$, and in the $2$ level, strength $2$ case it is isomorphic to $S_2^k\rtimes S_{k+1}$ for $k\geq 4$. The strength $2$ result reveals previously unknown permutation symmetries that can not be captured by the natural embedding of $S_2\wr S_k$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation., Comment: 17 pages

Statistical design of experiments is widely used in scientific and industrial investigations. A generalized minimum aberration (GMA) orthogonal array is optimum under the well-established, so-called GMA criterion, and such an array can extract as much information as possible at a fixed cost. Finding GMA arrays is an open (yet fundamental) problem in design of experiments because constructing such arrays becomes intractable as the number of runs and factors increase. We develop two directed enumeration algorithms that call the integer programming with isomorphism pruning algorithm of Margot (2007) for the purpose of finding GMA arrays. Our results include 16 GMA arrays that were not previously in the literature, along with documentation of the efficiencies that made the required calculations possible within a reasonable budget of computer time. We also validate heuristic algorithms against a GMA array catalog, by showing that they quickly output near GMA arrays, and then use the heuristics to find near GMA arrays when enumeration is computationally burdensome., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1501.02281

For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by $G^{\rm LP}$, is the symmetry group of the LP. Margot [F. Margot, 50 Years of Integer Programming 1958-2008 (2010), 647-686] described a method for computing a subgroup of the symmetry group $G^{\rm LP}$ of an LP. This method computes $G^{\rm LP}$ when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints. However, when the feasible set of the LP satisfies equality constraints this method finds only a subgroup of $G^{\rm LP}$ and can miss symmetries. We develop a method for finding the symmetry group of a feasible LP whose feasible set satisfies equality constraints. We apply this method to find and exploit the previously unexploited symmetries of an orthogonal array defining integer linear program (ILP) within the branch-and-bound (B\&B) with isomorphism pruning algorithm [F. Margot, Symmetric ILP: Coloring and small integers, Discrete Optimization 4 (1) (2007), 40-62]. Our method reduced the running time for finding all OD-equivalence classes of OA$(160,8,2,4)$ and OA$(176,8,2,4)$ by factors of $1/(2.16)$ and $1/(1.36)$ compared to the fastest known method [D. A. Bulutoglu and K. J. Ryan, Integer programming for classifying orthogonal arrays, Australasian Journal of Combinatorics 70 (3) (2018), 362-385]. These were the two bottleneck cases that could not have been solved until the B\&B with isomorphism pruning algorithm was applied., Comment: 27 pages

We provide a search method for Legendre pairs of composite length based on generating binary matrices with fixed row and column sums from compressed, complementary integer vectors. This approach yielded the first construction of a Legendre pair of length $77$, as well as the first exhaustive generation of Legendre pairs of length $55$., Comment: 17 pages, 1 figure. Submitted to Designs, Codes and Cryptography

By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points polytope of an orthogonal array (OA) defining integer linear program (ILP). Our results address the conjecture that if this polytope is non-empty, then it is full dimensional within the affine space where all the feasible points of the ILP's linear programming (LP) relaxation lie, raised by Appa et al., [On multi-index assignment polytopes, Linear Algebra and its Applications 416 (2-3) (2006), 224--241]. In particular, our theoretical results provide a sufficient condition for this polytope to be full dimensional within the LP relaxation affine space when it is non-empty. This sufficient condition implies all the known non-trivial values of the dimension of the $(k,s)$ assignment polytope. However, our results suggest that the conjecture mentioned above may not be true. More generally we provide previously unknown restrictions on the feasible values of the dimension of convex hull of all feasible points polytope of our OA defining ILP. We also determine all possible corresponding sets of equality constraints up to equivalence that can be implied by the integrality constraints of this ILP. Moreover, we find additional restrictions on the dimension of convex hull of feasible points and larger sets of corresponding equality constraints for the $n=2$ and even $s$ cases. These cases posses symmetries that do not necessarily exist in the $3\leq n$ or odd $s$ cases., Comment: 30 pages

Symmetry plays a fundamental role in design of experiments. In particular, symmetries of factorial designs that preserve their statistical properties are exploited to find designs with the best statistical properties. By using a result proved by Rosenberg [6], the concept of the LP relaxation orthogonal array polytope is developed and studied. A complete characterization of the permutation symmetry group of this polytope is made. Also, this characterization is verified computationally for many cases. Finally, a proof is provided., Comment: 12 pages

Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all non-isomorphic OA(N,k,s,t) given a set of all non-isomorphic OA(N,k-1,s,t). We also establish the concept of orthogonal design equivalence of OA(N,k,2,t) to reduce the number of integer linear programs (ILPs) whose all non-isomorphic solutions need to be enumerated by our methods. For each ILP, we determine the largest group of permutations that can be exploited with the branch-and-bound (B&B) with isomorphism pruning algorithm of Margot [Discrete Optim~4 (2007), 40-62] without losing isomorphism classes of OA(N,k,2,t). Our contributions brought the classifications of all non-isomorphic OA(160,k,2,4) for k=9,10 and OA(176,k,2,4) for k=5,6,7,8,9,10 within computational reach. These are the smallest s=2, t=4 cases for which classification results are not available in the literature., Comment: 21 pages

Booth and Cox proposed the E(s^2) criterion for constructing two-level supersaturated designs. Nguyen [Technometrics 38 (1996) 69-73] and Tang and Wu [Canad. J. Statist 25 (1997) 191-201] independently derived a lower bound for E(s^2). This lower bound can be achieved only when m is a multiple of N-1, where m is the number of factors and N is the run size. We present a method that uses difference families to construct designs that satisfy this lower bound. We also derive better lower bounds for the case where the Nguyen-Tang-Wu bound is not achievable. Our bounds cover more cases than a bound recently obtained by Butler, Mead, Eskridge and Gilmour [J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 621-632]. New E(s^2)-optimal designs are obtained by using a computer to search for designs that achieve the improved bounds., Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/009053604000000472