A group-based structure for perfect sequence covering arrays
An ( n , k )-perfect sequence covering array with multiplicity λ , denoted PSCA ( n , k , λ ) , is a multiset whose elements are permutations of the sequence ( 1 , 2 , ⋯ , n ) and which collectively contain each ordered length k subsequence exactly λ times. The primary objective is to determine for...
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| Published in | Designs, codes, and cryptography Vol. 91; no. 3; pp. 951 - 970 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.03.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0925-1022 1573-7586 |
| DOI | 10.1007/s10623-022-01132-1 |
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| Summary: | An (
n
,
k
)-perfect sequence covering array with multiplicity
λ
, denoted
PSCA
(
n
,
k
,
λ
)
, is a multiset whose elements are permutations of the sequence
(
1
,
2
,
⋯
,
n
)
and which collectively contain each ordered length
k
subsequence exactly
λ
times. The primary objective is to determine for each pair (
n
,
k
) the smallest value of
λ
, denoted
g
(
n
,
k
), for which a
PSCA
(
n
,
k
,
λ
)
exists; and more generally, the complete set of values
λ
for which a
PSCA
(
n
,
k
,
λ
)
exists. Yuster recently determined the first known value of
g
(
n
,
k
) greater than 1, namely
g
(
5
,
3
)
=
2
, and suggested that finding other such values would be challenging. We show that
g
(
6
,
3
)
=
g
(
7
,
3
)
=
2
, using a recursive search method inspired by an old algorithm due to Mathon. We then impose a group-based structure on a perfect sequence covering array by restricting it to be a union of distinct cosets of a prescribed nontrivial subgroup of the symmetric group
S
n
. This allows us to determine the new results that
g
(
7
,
4
)
=
2
and
g
(
7
,
5
)
∈
{
2
,
3
,
4
}
and
g
(
8
,
3
)
∈
{
2
,
3
}
and
g
(
9
,
3
)
∈
{
2
,
3
,
4
}
. We also show that, for each
(
n
,
k
)
∈
{
(
5
,
3
)
,
(
6
,
3
)
,
(
7
,
3
)
,
(
7
,
4
)
}
, there exists a
PSCA
(
n
,
k
,
λ
)
if and only if
λ
≥
2
; and that there exists a
PSCA
(
8
,
3
,
λ
)
if and only if
λ
≥
g
(
8
,
3
)
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0925-1022 1573-7586 |
| DOI: | 10.1007/s10623-022-01132-1 |