TY - JOUR
T1 - On Perfect Sequence Covering Arrays
AU - Gentle, Aidan R.
AU - Wanless, Ian M.
N1 - Funding Information:
The authors are grateful to Jingzhou Na, Jonathan Jedwab, and Shuxing Li for sharing the results of their ongoing investigation [, ], which has paralleled our own. We are also very grateful to Daniel Horsley who has been very generous with his time and advice. The first author was supported by an Australian Government Research Training Program (RTP) Scholarship. This research was supported by the Monash eResearch Centre through the use of the MonARCH HPC Cluster. Computations in Sect. were facilitated by GAP software [].
Publisher Copyright:
© 2022, The Author(s).
PY - 2023/9
Y1 - 2023/9
N2 - A PSCA(v, t, λ) is a multiset of permutations of the v-element alphabet { 0 , ⋯ , v- 1 } , such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly λ of the permutations. For v⩾ t⩾ 2 , we define g(v, t) to be the smallest positive integer λ, such that a PSCA(v, t, λ) exists. We show that g(6 , 3) = g(7 , 3) = g(7 , 4) = 2 and 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⩽ 32 and 3 ⩽ t⩽ 6. We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.
AB - A PSCA(v, t, λ) is a multiset of permutations of the v-element alphabet { 0 , ⋯ , v- 1 } , such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly λ of the permutations. For v⩾ t⩾ 2 , we define g(v, t) to be the smallest positive integer λ, such that a PSCA(v, t, λ) exists. We show that g(6 , 3) = g(7 , 3) = g(7 , 4) = 2 and 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⩽ 32 and 3 ⩽ t⩽ 6. We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.
KW - exact covering
KW - multiply transitive group
KW - permutation representation
KW - Sequence covering array
UR - http://www.scopus.com/inward/record.url?scp=85141034872&partnerID=8YFLogxK
U2 - 10.1007/s00026-022-00610-6
DO - 10.1007/s00026-022-00610-6
M3 - Article
AN - SCOPUS:85141034872
SN - 0218-0006
VL - 27
SP - 539
EP - 564
JO - Annals of Combinatorics
JF - Annals of Combinatorics
IS - 3
ER -