New covering array numbers

Balanced covering arrays: A classification of covering arrays and packing arrays via exact methods

“…If the column-wise linearization of P is lexicographically larger than the first s columns of A then the next symbol permutation for the last column is considered (lines 16 and 17); if all symbol permutations have been applied, then the next column of remaining is considered. Finally, if the partial array P is identical to the first s columns of A (indicating that the CA A could be a lex-leader) then in a recursive function call it is extended and all possible extensions of P are checked for lex-leadership (lines [20][21][22][23]. Once all columns in remaining have been examined, and no lexicographically smaller partial array was found, then A must be a lex-leader and we return TRUE (line 26).…”

“…| 253 additional λ vectors based on the bound in (26), which allows us for ≥ k 8 to consider only vectors ≽ λ (24, 12, 6, 2, 1). From Lemma 1 we know that for weaker λ vectors, there will be no new balanced CAs found with ≥ k 8: each 3-way interaction τ 3 must be covered at least 6 times, is a CAN number, see, for example [22] for verification. Nevertheless, we need to weaken λ to (26,12,6,2,1) (paired with y = (26,14,8,4,3)) to find the first optimal CA with 8 = (52;5,2) CAK columns, see Figure 19.…”

“…Besides this, we believe that the case of $${{\mathsf{CA}}}_{{\boldsymbol{\lambda }}}^{{\boldsymbol{y}}}(52,5,k,2)$$ is interesting because $$52={\mathsf{CAN}}(5,8,2)$$ is a CAN number, see, for example [22] for verification. Nevertheless, we need to weaken $${\boldsymbol{\lambda }}$$ to $$(26,12,6,2,1)$$ (paired with $${\boldsymbol{y}}=(26,14,8,4,3)$$) to find the first optimal CA with $$8={\mathsf{CAK}}(52;5,2)$$ columns, see Figure 19.…”