This paper concentrates on a class of pseudorandom sequences generated by combining q-ary m-sequences and quadratic characters over a finite field of odd order, called binary generalized NTU sequences. It is shown that the relationship among the sub-sequences of binary generalized NTU sequences can be formulated as combinatorial structures called Hadamard designs. As a consequence, the combinatorial structures generalize the group structure discovered by Kodera et al. (IEICE Trans. Fundamentals, vol.E102-A, no.12, pp.1659-1667, 2019) and lead to a finite-geometric explanation for the investigated group structure.

Combinatorial testing is an effective testing technique for detecting faults in a software or hardware system with multiple factors using combinatorial methods. By performing a test, which is an assignment of possible values to all the factors, and verifying whether the system functions as expected (pass) or not (fail), the presence of faults can be detected. The failures of the tests are possibly caused by combinations of multiple factors assigned with specific values, called faulty interactions. Martínez et al. [1] proposed the first deterministic adaptive algorithm for discovering faulty interactions involving at most two factors where each factor has two values, for which graph representations are adopted. In this paper, we improve Martínez et al.'s algorithm by an adaptive algorithmic approach for discovering faulty interactions in the so-called “non-2-locatable” graphs. We show that, for any system where each “non-2-locatable factor-component” involves two faulty interactions (for example, a system having at most two faulty interactions), our improved algorithm efficiently discovers all the faulty interactions with an extremely low mistaken probability caused by the random selection process in Martínez et al.'s algorithm. The effectiveness of our improved algorithm are revealed by both theoretical discussions and experimental evaluations.

In this paper, we propose a notion for high-dimensional generalizations of mutually orthogonal Latin squares (MOLS) and mutually orthogonal diagonal Latin squares (MODLS), called mutually dimensionally orthogonal d-cubes (MOC) and mutually dimensionally orthogonal diagonal d-cubes (MODC). Systematic constructions for MOC and MODC by using polynomials over finite fields are investigated. In particular, for 3-dimensional cubes, the results for the maximum possible number of MODC are improved by adopting the proposed construction.