Perfect sequence covering arrays

Yuster, Raphael

doi:10.1007/s10623-019-00698-7

Cited by 13 publications

(28 citation statements)

References 15 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But, as mentioned earlier, even for the smallest case of r = 2, namely graphs, we do not know whether there exists a n n λ 4 − ( , , ) directed design in which λ is polynomial in n. So, we cannot use n n λ 4 − ( , , ) directed designs to solve Problem 1.2 for graphs. The situation is even worse since it is known that in any n n λ 4 − ( , , ) directed design, λ is at least quadratic in n [21], so 788 | YUSTER we will never be able to use n n λ 4 − ( , , ) directed designs to obtain Theorem 1.3. On the other hand, n n λ 4 − ( , , ) directed designs and PSD K ( ) n seem "so close."…”

Section: Yuster | 787mentioning

confidence: 99%

“…Fairly close lower and upper bounds are known for the minimum size of a

k

‐sequence covering array of

[n]

, the state of the art for the case

k = 3

given in [12] (lower bound), [20] (upper bound) and for general

k

given in [17] (lower bound), [19] (upper bound). Fairly close upper and lower bounds are known for the minimum size of a perfect 3‐sequence covering array of

[n]

[21]. Trivially, every perfect 4‐sequence covering array is a perfect separating family.…”

Section: Constructing a Perfect Separating Familymentioning

confidence: 99%

“…Trivially, every perfect 4‐sequence covering array is a perfect separating family. Unfortunately, as we have already mentioned in the introduction, a lower bound of

normalΩ (n^{2})

is known for the minimum size of a perfect 4‐sequence covering array [21]. Since Theorem 1.3 asserts a quasi‐linear upper bound for a perfect separating family, we cannot use perfect 4‐sequence covering arrays for our construction.…”

Section: Constructing a Perfect Separating Familymentioning

confidence: 99%

“…Unfortunately, we do not yet know much about that minimum

λ

. In fact, for all

t \geq 4

, no polynomial lower bound for

λ

is known [21]. As we shall see below, this particular difficulty (already at

t = 4

) is one of the most intriguing reasons to look at the design‐theoretic counterpart of separation dimension (which we call perfect separation dimension below) in addition to it being a very natural problem.…”

Section: Introductionmentioning

confidence: 99%

“…So, we cannot use

4 - (n, n, λ)

directed designs to solve Problem 1.2 for graphs. The situation is even worse since it is known that in any

4 - (n, n, λ)

directed design,

λ

is at least quadratic in

n

[21], so we will never be able to use

4 - (n, n, λ)

directed designs to obtain Theorem 1.3. On the other hand,

4 - (n, n, λ)

directed designs and

P S D (K_{n})

seem “so close.” Indeed, in the former, we need to satisfy

n (n - 1) (n - 2) (n - 3) = normalΘ (n^{4})

requirements as these are the number of sequences of four distinct elements of

n

while in the latter, we need to satisfy

n (n - 1) (n - 2) (n - 3) ∕ 8 = normalΘ (n^{4})

requirements as these are the number of disjoint pairs of edges of

K_{n}

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Perfect and nearly perfect separation dimension of complete and random graphs

Yuster

2021

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Yuster | 787mentioning

confidence: 99%

“…Fairly close lower and upper bounds are known for the minimum size of a

k

‐sequence covering array of

[n]

, the state of the art for the case

k = 3

given in [12] (lower bound), [20] (upper bound) and for general

k

given in [17] (lower bound), [19] (upper bound). Fairly close upper and lower bounds are known for the minimum size of a perfect 3‐sequence covering array of

[n]

[21]. Trivially, every perfect 4‐sequence covering array is a perfect separating family.…”

Section: Constructing a Perfect Separating Familymentioning

confidence: 99%

“…Trivially, every perfect 4‐sequence covering array is a perfect separating family. Unfortunately, as we have already mentioned in the introduction, a lower bound of

normalΩ (n^{2})

Section: Constructing a Perfect Separating Familymentioning

confidence: 99%

“…Unfortunately, we do not yet know much about that minimum

λ

. In fact, for all

t \geq 4

, no polynomial lower bound for

λ

is known [21]. As we shall see below, this particular difficulty (already at

t = 4

Section: Introductionmentioning

confidence: 99%

“…So, we cannot use

4 - (n, n, λ)

directed designs to solve Problem 1.2 for graphs. The situation is even worse since it is known that in any

4 - (n, n, λ)

directed design,

λ

is at least quadratic in

n

[21], so we will never be able to use

4 - (n, n, λ)

directed designs to obtain Theorem 1.3. On the other hand,

4 - (n, n, λ)

directed designs and

P S D (K_{n})

seem “so close.” Indeed, in the former, we need to satisfy

n (n - 1) (n - 2) (n - 3) = normalΘ (n^{4})

requirements as these are the number of sequences of four distinct elements of

n

while in the latter, we need to satisfy

n (n - 1) (n - 2) (n - 3) ∕ 8 = normalΘ (n^{4})

requirements as these are the number of disjoint pairs of edges of

K_{n}

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Perfect and nearly perfect separation dimension of complete and random graphs

Yuster

2021

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

show abstract

On Perfect Sequence Covering Arrays

Gentle

Wanless

2022

Ann. Comb.

View full text Add to dashboard Cite

A PSCA$$(v, t, \lambda )$$ ( v , t , λ ) is a multiset of permutations of the v-element alphabet $$\{0, \dots , v-1\}$$ { 0 , ⋯ , v - 1 } , such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly $$\lambda $$ λ of the permutations. For $$v \geqslant t \geqslant 2$$ v ⩾ t ⩾ 2 , we define g(v, t) to be the smallest positive integer $$\lambda $$ λ , such that a PSCA$$(v, t, \lambda )$$ ( v , t , λ ) exists. We show that $$g(6, 3) = g(7, 3) = g(7, 4) = 2$$ g ( 6 , 3 ) = g ( 7 , 3 ) = g ( 7 , 4 ) = 2 and $$g(8, 3) = 3$$ g ( 8 , 3 ) = 3 . Using suitable permutation representations of groups, we make improvements to the upper bounds on g(v, t) for many values of $$v \leqslant 32$$ v ⩽ 32 and $$3\leqslant t\leqslant 6$$ 3 ⩽ t ⩽ 6 . We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.

show abstract