On E(s2)‐optimal and minimax‐optimal supersaturated designs with 20 rows and 76 columns

Morales, Luis B.; Bulutoglu, Dursun A.

doi:10.1002/jcd.21595

Cited by 2 publications

(13 citation statements)

References 37 publications

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“…Let

Q goodbreakinfix= (B_{1}, B_{2})

and

Q^{'} goodbreakinfix= (B_{1}^{'}, B_{2}^{'})

be two different parallel classes of a RIBD

(n, m, n ∕ 2)

. Define their parallel class intersection matrix (PCIM) as the

2 goodbreakinfix\times 2

matrix

bold-italicA (Q, Q^{'})

with entries defined by

a_{i j} goodbreakinfix= ∣ B_{i} goodbreakinfix\cap B_{j}^{'} ∣

. A RIBD with parameters

(n, m, n ∕ 2)

has

m ∕ 2

parallel classes.…”

Section: Ssds and Resolvable Ribdsmentioning

confidence: 99%

“…In consequence, our isomorph rejection was based on all isomorphisms that left the sets

{1, 2, 3}

H goodbreakinfix- {1, 2, 3}

, and

{n ∕ 2 + 2 ℓ, n ∕ 2 + 2 ℓ + 1}

for

0 goodbreakinfix\leq ℓ goodbreakinfix\leq (n - 2) ∕ 4

invariant. This partial isomorph rejection scheme drastically reduced the computer time required to search for a clique of size

12

as in .…”

Section: The Algorithm For Ngoodbreakinfix=22mentioning

confidence: 99%

“…By Lemma 9, we assumed WLOG that Q 1 of the RIBD D E ( ) was of type 1 and equal to ({1, 2, 3, 5, 6}, {4, 7, 8, 9, 10}). By relabeling the blocks in the parallel classes if is necessary, each of the 11 matrices associated with Q 1 is assumed to be as in (10) or (11). Then, by Lemma 9 and Table 2(a) we could choose the other 11 parallel classes in…”

Section: Cases N = 22 and N = 26mentioning

confidence: 99%

“…(2, 6, 9) = 5, (11,6,9) = 5, (12, 6, 9) = 5, (2, 11, 12) = 0. However, an exhaustive search shows that there exists no nonnegative integer solution to constraints (28) and (29).…”

Section: The Algorithm For N = 22mentioning

confidence: 99%

“…V n = {1,…, }, and let be the set of all pairs B B ( , ) such that B B { , } is a partition of V with | | ∕ B n =(i = 1, 2). We now construct a graph with one vertex for each element in , where an edge connects two vertices Q B whenever the PCIM of Q and Q ′ has the form (10) or(11). Then, by(10) or(11) and Theorem 1, the search of an SSD, D n n ( , + 2), with ≤ s is equivalent to finding a clique of size n + 2 in the graph .…”

mentioning

confidence: 99%

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The maximum number of columns in supersaturated designs with

Morales

Bulutoglu

Arasu

2019

J of Combinatorial Designs

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Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns B ( n , t ) that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows ( n) and a maximum in absolute value correlation between any two columns ( t ∕ n). In particular, they proved that B ( n , 2 ) goodbreakinfix≤ n goodbreakinfix+ 2 for n goodbreakinfix≡ 2 (mod 4) and n goodbreakinfix> 6. However, the only known SSD satisfying this upper bound is when n goodbreakinfix= 10. By utilizing a computer search, we prove that B ( n , 2 ) goodbreakinfix≤ n goodbreakinfix+ 1 for n goodbreakinfix= 18 goodbreakinfix, 22 goodbreakinfix, 30, and B ( 14 , 2 ) goodbreakinfix= 15. These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the E ( s 2 ) lower bound for SSDs with n rows and n goodbreakinfix+ 2 columns, for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22, and 30. Finally, we show that a skew‐type Hadamard matrix of order n can be used to construct an SSD with n goodbreakinfix− 2 rows and n goodbreakinfix− 1 columns that proves B ( n − 2 , 2 ) goodbreakinfix≥ n goodbreakinfix− 1. Hence, we establish B ( n , 2 ) goodbreakinfix= n goodbreakinfix+ 1 for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22 goodbreakinfix, 30 and B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 for all n goodbreakinfix≡ 2 (mod 4) such that n goodbreakinfix≤ 270. Our result also implies that B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 when n goodbreakinfix+ 1 is a prime power and n goodbreakinfix+ 1 goodbreakinfix≡ 3 (mod 4). We conjecture that n goodbreakinfix+ 1 goodbreakinfix= B ( n , 2 ) goodbreakinfix 10 and n goodbreakinfix≡ 2 (mod 4), where B ′ ( n , 2 ) is the maximum number of equiangular lines in R n − 1 with pairwise angle arccos ( 2 ∕ n ).

show abstract

“…Let

Q goodbreakinfix= (B_{1}, B_{2})

and

Q^{'} goodbreakinfix= (B_{1}^{'}, B_{2}^{'})

be two different parallel classes of a RIBD

(n, m, n ∕ 2)

. Define their parallel class intersection matrix (PCIM) as the

2 goodbreakinfix\times 2

matrix

bold-italicA (Q, Q^{'})

with entries defined by

a_{i j} goodbreakinfix= ∣ B_{i} goodbreakinfix\cap B_{j}^{'} ∣

. A RIBD with parameters

(n, m, n ∕ 2)

has

m ∕ 2

parallel classes.…”

Section: Ssds and Resolvable Ribdsmentioning

confidence: 99%

“…In consequence, our isomorph rejection was based on all isomorphisms that left the sets

{1, 2, 3}

H goodbreakinfix- {1, 2, 3}

, and

{n ∕ 2 + 2 ℓ, n ∕ 2 + 2 ℓ + 1}

for

0 goodbreakinfix\leq ℓ goodbreakinfix\leq (n - 2) ∕ 4

invariant. This partial isomorph rejection scheme drastically reduced the computer time required to search for a clique of size

12

as in .…”

Section: The Algorithm For Ngoodbreakinfix=22mentioning

confidence: 99%

Section: Cases N = 22 and N = 26mentioning

confidence: 99%

“…(2, 6, 9) = 5, (11,6,9) = 5, (12, 6, 9) = 5, (2, 11, 12) = 0. However, an exhaustive search shows that there exists no nonnegative integer solution to constraints (28) and (29).…”

Section: The Algorithm For N = 22mentioning

confidence: 99%

mentioning

confidence: 99%

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The maximum number of columns in supersaturated designs with

Morales

Bulutoglu

Arasu

2019

J of Combinatorial Designs

Self Cite

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The maximum number of columns in E(s2) $\,E({s}^{2})$‐optimal supersaturated designs with 16 rows and smax=4 ${s}_{{\rm{\max }}}=4$ is 60

Morales

2023

J of Combinatorial Designs

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We show that the maximum number of columns in 0.1emE(s2) $\,E({s}^{2})$‐optimal supersaturated designs (SSDs) with 16 rows and smax=4 ${s}_{{\rm{\max }}}=4$ is 60 by showing that there exists no resolvable 2‐(16, 8, 35) design such that any two blocks from different parallel classes intersect in 3, 5, or 4 points. This is accomplished by an exhaustive computer search that uses the parallel class intersection pattern method to reduce the search space. We also classify all nonisomorphic 0.1emE(s2) $\,E({s}^{2})$‐optimal 5‐circulant SSDs with 16 rows and smax=8 ${s}_{{\rm{\max }}}=8$.

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On E(s²)‐optimal and minimax‐optimal supersaturated designs with 20 rows and 76 columns

Cited by 2 publications

References 37 publications

The maximum number of columns in supersaturated designs with

The maximum number of columns in supersaturated designs with

The maximum number of columns in E(s2) $\,E({s}^{2})$‐optimal supersaturated designs with 16 rows and smax=4 ${s}_{{\rm{\max }}}=4$ is 60

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