Construction of group strong orthogonal arrays of strength two plus

Abstract

Strong orthogonal arrays (SOAs) have received more and more attention recently since they enjoy more desirable space-filling properties than ordinary orthogonal arrays. Among them, the SOAs of strength $2+$ are the most advisable as they satisfy the same two-dimensional space-filling property as SOAs of strength 3 while having more columns for given run sizes. In addition, column-orthogonality is also a desirable property for designs of computer experiments. Existing column-orthogonal SOAs of strength $2+$ have limited columns. In this paper, we propose a new class of space-filling designs, called group SOAs of strength $2+$, and provide construction methods for such designs. The proposed designs can accommodate more columns than column-orthogonal SOAs of strength $2+$ for given run sizes while satisfying similar stratifications and retaining a high proportion of column-orthogonal columns. Orthogonal arrays and difference schemes play important roles in the construction. The construction procedures are easy to implement and a large amount of group SOAs with $s^2$ levels are constructed where $s \ge 2$ is a prime power. In addition, the run sizes of the constructed designs are s times the ones of the orthogonal arrays used in the construction procedure. Thus they are relatively flexible.

Appendices

Appendix

A.1 Some multiplication tables of GF(s)

See Tables 9, 10, and 11.

Table 9 Multiplication table of GF(3)

Table 10 Multiplication table of GF(4)

Table 11 Multiplication table of GF(5)

A.2 Proofs of theorems

Proof of Theorem 1   First, in order to prove that the constructed design D can be collapsed into an OA(2n, u, 2, 2), we need to prove that A is an OA of strength 2. As the Kronecker sum with modulo 2 of an OA of strength 2 and the following matrix

$$\begin{aligned}\begin{pmatrix} 0&{}\ 0\\ 0&{}\ 1\\ \end{pmatrix}\end{aligned}$$

is still an OA of strength 2, then A is an OA of strength 2.

To prove that any two distinct columns from the same group can be collapsed into an OA$(2n,2,4\times 2, 2)$ and an OA$(2n,2,2\times 4, 2)$, we need further prove that $(a_{kq}, b_{kq}, a_{kl})$ is an OA of strength 3, $1\le k\le \lfloor m/2\rfloor +1$, $q\ne l$. For $k=1$, $(a_{1q}, b_{1q}, a_{1l})$ can be written as

$$\begin{aligned} \begin{pmatrix}c_q&{}c_q+1_n&{}c_l\\ c_q&{}c_q&{}c_l\end{pmatrix}. \end{aligned}$$

As $c_q$ is a two-level vector with entries 0 and 1, $c_q+1_n$ can be obtained from $c_q$ by replacing entry 0 by 1, and 1 by 0. It is easy to see that the array is an OA of strength 3. For $k=2, \ldots , \lfloor m/2\rfloor +1$,

$$\begin{aligned} (a_{k1}, b_{k1}, a_{k2})=\begin{pmatrix}c_{k-1}&{}c_{m-k+2}+1_n&{}c_{m-k+2}\\ c_{k-1}+1_n&{}c_{m-k+2}+1_n&{}c_{m-k+2}+1_n\end{pmatrix}. \end{aligned}$$

Due to the property of $c_i$, we can obtain that it is an OA of strength 3. For $(a_{k2}, b_{k2}, a_{k1})$, the proof is similar.

To prove (i) of Theorem 1, we need to prove $(a_{1q}, b_{1q}, a_{kl})$ and $(a_{kl}, b_{kl}, a_{1q})$ are OAs of strength 3 when $q\ne k-1$ and $m-k+2$. $(a_{1q}, b_{1q}, a_{kl})$ and $(a_{kl}, b_{kl}, a_{1q})$ have the following expressions:

$$\begin{aligned}\begin{pmatrix} c_i&{}{}c_i+1_n&{}{}c_j\\ c_i&{}{}c_i&{}{}c_j+1_n \end{pmatrix} \text{ with } i\ne j, \text{ and } \quad \begin{pmatrix} c_i&{}{}c_j+1_n&{}{}c_k\\ c_i&{}{}c_j+1_n&{}{}c_k \end{pmatrix} \text{ with } i\ne j\ne k.\end{aligned}$$

As $c_i$, $c_j$ and $c_k$ are columns of OA of strength 2, we can obtain that both expressions are OAs of strength 3. Then for an odd m, the total number of pairs of columns that can achieve stratifications on $4 \times 2$ and $2\times 4$ grids is at least $(3m-3)(m-1)/2$, so $p_1\ge (3m-3)/(4m-2)$; and for an even m, it is at least $(3m-4)m/2$, thus $p_1\ge (3m-4)/(4m-2)$.

To prove (ii) of Theorem 1, without loss of generality, we assume that the levels of A and B have been centered. Then the inner product of any two columns of D $d_{pq}^Td_{kl}=4a_{pq}^Ta_{kl}+2a_{pq}^Tb_{kl}+2b_{pq}^Ta_{kl}+b_{pq}^Tb_{kl}$, where $p\ne k$ or $q\ne l$. It is equal to zero if and only if ${a_{pq}}$ and $a_{kl}$, ${a_{pq}}$ and $b_{kl}$, ${b_{pq}}$ and $a_{kl}$, ${b_{pq}}$ and $b_{kl}$ are column-orthogonal. It is clear that $a_{pq}^Ta_{kl}=b_{pq}^Tb_{kl}=0$. For the other two terms, without loss of generality, we assume $p<q$ when $p\ne q$. There are three cases:

• (1) when $p=1$ and $q=k-1$, $a_{pq}^Tb_{kl}=-2n$, $b_{pq}^Ta_{kl}=0$, then $\rho (d_{pq}, d_{kl})=-4n/10n=-0.4$;

• (2) when $p=1$ and $q=m-k+2$, $a_{pq}^Tb_{kl}=0$, $b_{pq}^Ta_{kl}=-2n$, then $\rho (d_{pq}, d_{kl})=-4n/10n=-0.4$;

• (3) for all other conditions, $a_{pq}^Tb_{kl}=b_{pq}^Ta_{kl}=0$.

Thus, we can get $p_2=(2m-3)/(2m-1)$. The proof is completed.

Proof of Theorem 2.   To prove that the constructed design can be collapsed into an OA(sn, u, s, 2), we need to prove that A is an OA of strength 2. Note that any two distinct columns of A is

$$\begin{aligned} (a_{pq}, a_{kl})=\begin{pmatrix} c_i^l&{}c_{j}^{l'}\\ c_{i}^l+1_n&{}c_{j}^{l'}+1_n\\ \vdots &{}\vdots \\ c_{i}^l+(s-1)1_n&{}c_{j}^{l'}+(s-1)1_n\\ \end{pmatrix}. \end{aligned}$$

This form can be divided into two cases:

• (1) when $i\ne j$, then $(c_{i}^l, c_{j}^{l'})$ is an OA, $(a_{pq}, a_{kl})$ is s times replication of it, so it is also an OA;

• (2) when $i= j$, then $(c_{i}^l, c_{j}^{l'})=(c_{i}^l, c_{i}^{l'})$, as $c_{i}^l$ and $c_{i}^{l'}$ are obtained from $c_{i}$ by replacing the levels with the elements of the difference scheme H, $(c_{i}^l, c_{i}^{l'})$ is a difference scheme. For any $\sigma ,\sigma '\in GF(s)$, we must show that the number of runs $(\sigma , \sigma ')$ is equal to n/s. We know that n/s entries in $c_{i}^l-c_{i}^{l'}$ are equal to $\sigma -\sigma '$. Without loss of generality, we assume that the run $(\sigma , \sigma ')$ occurs in $(c_{i}^l, c_{i}^{l'})$, then there is a unique row $(\sigma , \sigma ')$ for each occurrence of $\sigma -\sigma '$ in $c_{i}^l-c_{i}^{l'}$, and the row will not occur in $(c_{i}^l+j1_n, c_{i}^{l'}+j1_n)$, $j=2,\ldots ,s-1$, hence $(a_{pq}, a_{kl})$ is an OA.

To prove that any two distinct columns $d_{kp}$ and $d_{kq}$ from the kth group can be collapsed into an OA$(sn,2,s^2\times s, 2)$ and an OA$(sn,2,s\times s^2, 2)$, we need to prove that $(a_{kp}, b_{kp}, a_{kq})$ and $(a_{kq}, b_{kq}, a_{kp})$ are both OAs of strength 3. Let us first consider $(a_{kp}, b_{kp}, a_{kq})$ which has three cases.

Case 1:

$$\begin{aligned} (a_{kp}, b_{kp}, a_{kq})= \begin{pmatrix} c_k^l&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l'}\\ c_k^l+1_n&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l'}+1_n\\ \vdots &{}\vdots &{}\vdots \\ c_k^l+(s-1)1_n&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l'}+(s-1)1_n\\ \end{pmatrix}. \end{aligned}$$

As $(c_k^l, c_{m+1-k}^{l'})$ is an OA, it consists of the $s^2$ different 2-tuples with $n/s^2$ times each. It is obvious that $c_{m+1-k}^{l'}$ is a level permutation of $c_{m+1-k}^{l}$, hence $(c_k^l, c_{m+1-k}^{l}, c_{m+1-k}^{l'})$ consists of the $s^2$ different 3-tuples $n/s^2$ times each. It is similar for $(c_k^l+j1_n, c_{m+1-k}^{l}, c_{m+1-k}^{l'}+j1_n)$, $j\in \{1,\ldots ,s-1\}$. For the same reason, $(c_{m+1-k}^{l}, c_{m+1-k}^{l'}+i1_n)$ and $(c_{m+1-k}^{l}, c_{m+1-k}^{l'}+j1_n)$ consist of different 2-tuples when $i\ne j$, where $i,j \in GF(s)$. Hence there are $s^3$ different 3-tuples in $(a_{kp}, b_{kp}, a_{kq})$ with $n/s^2$ times each, $(a_{kp}, b_{kp}, a_{kq})$ is an OA of strength 3.

Case 2:

$$\begin{aligned} (a_{kp}, b_{kp}, a_{kq})= \begin{pmatrix} c_k^l&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l}\\ c_k^l+1_n&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l}+1_n\\ \vdots &{}\vdots &{}\vdots \\ c_k^l+(s-1)1_n&{}c_{m+1-k}^{l}&{}c_{m+1-k}^{l}+(s-1)1_n\\ \end{pmatrix}. \end{aligned}$$

As $c_{m+1-k}^{l}$ is a level permutation of itself, this is a special case of Case 1.

Case 3:

$$\begin{aligned} (a_{kp}, b_{kp}, a_{kq})= \begin{pmatrix} c_k^l&{}c_{m+1-k}^{l}&{}c_{k}^{l'}\\ c_k^l+1_n&{}c_{m+1-k}^{l}&{}c_{k}^{l'}+1_n\\ \vdots &{}\vdots &{}\vdots \\ c_k^l+(s-1)1_n&{}c_{m+1-k}^{l}&{}c_{k}^{l'}+(s-1)1_n\\ \end{pmatrix}. \end{aligned}$$

It is obvious that $(c_k^l, c_k^{l'})$ consists of the s different 2-tuples n/s times each. As $(c_k^{l}, c_{m+1-k}^{l})$ is an OA, $(c_k^l, c_{m+1-k}^{l}, c_{k}^{l'})$ consists of the $s^2$ different 3-tuples with $n/s^2$ times each. It is similar for $(c_k^l+j 1_n, c_{m+1-k}^{l}, c_{k}^{l'}+j 1_n)$, $j\in \{1,\ldots ,s-1\}$. As $(c_k^l, c_k^{l'})$ is a difference scheme, $(c_k^l+ i 1_n, c_k^{l'}+i 1_n)$ and $(c_k^l+ j 1_n, c_k^{l'}+j 1_n)$ consist of different combinations when $i\ne j$, where $i,j \in GF(s)$. Hence there are $s^3$ different 3-tuples in $(a_{kp}, b_{kp}, a_{kq})$ with $n/s^2$ times each, $(a_{kp}, b_{kp}, a_{kq})$ is an OA of strength 3.

For $(a_{kq}, b_{kq}, a_{kp})$, it is similar. Thus, we conclude that any two distinct columns from the same group can be collapsed into an OA$(sn,2,s^2\times s, 2)$ and an OA$(sn,2,s\times s^2, 2)$ now.

For (i) and (ii) of Theorem 2, when m is even, $\lceil m/2\rceil =m/2$, the resulting design is a GSOA$(ns, (s^2)^{2e\cdot (m/2)},2+)$ with the total number of columns being me, the number of column pairs achieving stratifications on $s^2\times s$ and $s\times s^2$ grids is at least $me(2e-1)/2$, so $p_1\ge (2e-1)/(me-1)$; the number of column-orthogonal column pairs is $me^2(m-1)/2$, then $p_2=e(m-1)/(me-1)$. When m is odd, $\lceil m/2\rceil =(m+1)/2$, the resulting design is a GSOA$(ns, (s^2)^{2e\cdot (m-1)/2+1},2+)$ with the total number of columns being $(m-1)e+1$, the number of column pairs achieving stratifications on $s^2\times s$ and $s\times s^2$ grids is at least $e(2e-1)(m-1)/2$, so $p_1\ge (2e-1)/(me-e+1)$; the number of column-orthogonal column pairs is $e(m-1)(me-2e+2)/2$, then $p_2=(me-2e+2)/(me-e+1)$.

The proof of (iii) of Theorem 2 is similar to that of Theorem 1. Any two columns of design D, ${d_{pq}}$ and $d_{kl}$ are column-orthogonal if and only if ${a_{pq}}$ and $a_{kl}$, ${a_{pq}}$ and $b_{kl}$, ${b_{pq}}$ and $a_{kl}$, ${b_{pq}}$ and $b_{kl}$ are column-orthogonal. Without loss of generality, we assume that the levels of A and B have been centered. It is easy to see, $a_{pq}^Ta_{kl}=a_{pq}^Tb_{kl}=b_{pq}^Ta_{kl}=0$. For $b_{kl}^T{b_{pq}}$, there are two cases:

• (1) $b_{kl}^T{b_{pq}}=0$ when $p\ne k$;

• (2) when $p=k$, without loss of generality, we assume $q<l$. For this case,

  $$\begin{aligned} b_{kl}^T{b_{kq}}= {\left\{ \begin{array}{ll} \ 0, &{} {\text{ if } 1\le q\le e,\ e+1\le l\le 2e}\\ nv_{ql},&{} {\text{ if } 1\le q<l \le e \text{ or } e+1\le q< l\le 2e,} \end{array}\right. } \end{aligned}$$

  (1)

  where $v_{ql}=\langle h_{q({mod}\ e)},h_{l({mod}\ e)}\rangle$. So the correlation of $d_{kq}$ and $d_{kl}$ is

  $$\begin{aligned} \rho _{(d_{kq},d_{kl})}= {\left\{ \begin{array}{ll} 12v_{ql}/f,&{} {\text{ if } s \text{ is } \text{ odd },}\\ \ \ 3v_{ql}/f,&{} {\text{ if } s \text{ is } \text{ even },} \end{array}\right. } \end{aligned}$$

  (2)

  where $f=s(s^2-1)(s^2+1)$.

The proof is completed.