Abstract A Steiner triple system STS( v ) is called f -pyramidal if it has an automorphism group fixing f points and acting sharply transitively on the remaining $$v-f$$ v - f points. In this paper, we focus on the STSs that are f -pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of f , that is, $$f=0,1,3$$ f = 0 , 1 , 3 . In this paper, we complete this result and determine, for every $$f>3$$ f > 3 , the spectrum of values ( f , v ) for which there is an f -pyramidal STS( v ) over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes$\mathcal {C}_{(q,q+1,4,1)}$and their dual codes with$q=2^{m}$and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of$t$-designs with$t=3,4$. In particular, we prove that codewords of weight 7 in$\mathcal {C}_{(q,q+1,4,1)}$support 4-designs for odd$m \geqslant 5$and they support 3-designs for even$m \geqslant 4$, which provide infinite classes of simple$t$-designs with new parameters. Another significant class of$t$-designs we produce in this paper has complementary designs with parameters 4-$(2^{2s+1}+ 1,5,5)$; these designs have the smallest index among all the known simple 4-$(q+1,5,\lambda)$designs derived from codes for prime powers$q$; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL$(2,2^{2s+1})$as the automorphism group constructed by Alltop in 1969.

Sequential processes can encounter faults as a result of improper ordering of subsets of the events. In order to reveal faults caused by the relative ordering of $t$ or fewer of $v$ events, for some fixed $t$, a test suite must provide tests so that every ordering of every set of $t$ or fewer events is exercised. Such a test suite is equivalent to a sequence covering array, a set of permutations on $v$ events for which every subsequence of $t$ or fewer events arises in at least one of the permutations. Equivalently it is a (different) set of permutations, a completely $t$-scrambling set of permutations, in which the images of every set of $t$ chosen events include each of the $t!$ possible “patterns.” In event sequence testing, minimizing the number of permutations used is the principal objective. By developing a connection with covering arrays, lower bounds on this minimum in terms of the minimum number of rows in covering arrays are obtained. An existing bound on the largest $v$ for which the minimum can equal $t!$ is improved. A conditional expectation algorithm is developed to generate sequence covering arrays whose number of permutations never exceeds a specified logarithmic function of $v$ when $t$ is fixed, and this method is shown to operate in polynomial time. A recursive product construction is established when $t=3$ to construct sequence covering arrays on $vw$ events from ones on $v$ and $w$ events. Finally computational results are given for $t \in \{3,4,5\}$ to demonstrate the utility of the conditional expectation algorithm and the product construction.