A group-based structure for perfect sequence covering arrays - Na, Jingzhou; Jedwab, Jonathan; Li, Shuxing (Designs, codes, and cryptography, 2023)
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 ) .