Covering arrays with row limit, CARLs, are a new family of combinatorial objects which we introduce as a generalization of group divisible designs and covering arrays. In the same manner as their predecessors, CARLs have a natural application as combinatorial models for interaction test suites. A CARL(N;t,k,v:w), is an N×k array with some empty cells. A component, which is represented by a column, takes values from a v-set called the alphabet. In each row, there are exactly w non-empty cells, that is the corresponding components have an assigned value from the alphabet. The parameter w is called the row limit. Moreover, any N×t subarray contains every of v^t distinct t-tuples of alphabet symbols at least once. This thesis is concerned with the bounds on the size and with the construction of CARLs when the row limit w(k) is a positive integer valued function of the number of columns, k. Here we give a lower bound, and probabilistic and algorithmic upper bounds for any CARL. Further, we find improvements on the upper bounds when w(k)ln(w(k)) = o(k) and when w(k) is a constant function. We also determine the asymptotic size of CARLs when w(k) = Θ(k) and when w(k) is constant. Next, we study constructions of CARLs. We provide two combinatorial constructions of CARLs, which we apply to construct families of CARLs with w(k)=ck, where c<1. Also, we construct optimal CARLs when t=2 and w=4, and prove that there exists a constant δ, such that for any v and k≥4, an optimal CARL(2,k,v:4) differs from the lower bound by at most δ rows, with some possible exceptions. Finally, we define a packing array with row limit, PARL(N;t,k,v:w), in the same way as a CARL(N;t,k,v:w) with the difference that any t-tuple is contained at most once in any N×t subarray. We find that when w(k) is a constant function, the results on the asymptotic size of CARLs imply the results on the asymptotic size of PARLs. Also, when t=2, we consider a transformation of optimal CARLs with row limit w=3 to optimal PARLs with w=3.

Covering arrays with row limit, CARLs, are a new family of combinatorial objects which we introduce as a generalization of group divisible designs and covering arrays. In the same manner as their predecessors, CARLs have a natural application as combinatorial models for interaction test suites. A CARL(N;t,k,v:w), is an N×k array with some empty cells. A component, which is represented by a column, takes values from a v-set called the alphabet. In each row, there are exactly w non-empty cells, that is the corresponding components have an assigned value from the alphabet. The parameter w is called the row limit. Moreover, any N×t subarray contains every of v^t distinct t-tuples of alphabet symbols at least once. This thesis is concerned with the bounds on the size and with the construction of CARLs when the row limit w(k) is a positive integer valued function of the number of columns, k. Here we give a lower bound, and probabilistic and algorithmic upper bounds for any CARL. Further, we find improvements on the upper bounds when w(k)ln(w(k)) = o(k) and when w(k) is a constant function. We also determine the asymptotic size of CARLs when w(k) = Θ(k) and when w(k) is constant. Next, we study constructions of CARLs. We provide two combinatorial constructions of CARLs, which we apply to construct families of CARLs with w(k)=ck, where c<1. Also, we construct optimal CARLs when t=2 and w=4, and prove that there exists a constant δ, such that for any v and k≥4, an optimal CARL(2,k,v:4) differs from the lower bound by at most δ rows, with some possible exceptions. Finally, we define a packing array with row limit, PARL(N;t,k,v:w), in the same way as a CARL(N;t,k,v:w) with the difference that any t-tuple is contained at most once in any N×t subarray. We find that when w(k) is a constant function, the results on the asymptotic size of CARLs imply the results on the asymptotic size of PARLs. Also, when t=2, we consider a transformation of optimal CARLs with row limit w=3 to optimal PARLs with w=3.

Combinatorial Group Testing (CGT) is a process of identifying faulty interactions (“errors”) within a particular set of items. Error Locating Arrays (ELAs) are combinatorial designs that can be built from Covering Arrays (CAs) to not only cover all errors in a system (each involving up to a certain number of items), but to locate and identify the errors as well. In this thesis, we survey known results for CGT, as well as CAs, ELAs, and some other types of related arrays. More importantly, we give several new results. First, we give a new algorithm that can be used to test a system in which each component (factor) has two options (values), and at most two errors are present. We show that, for systems with at most two errors, our algorithm improves upon a related algorithm by Mart´ınez et al. in terms of both robustness and efficiency. Second, we give the first adaptive CGT algorithm that can identify, among a given set of k items, all faulty interactions involving up to three items. We then compare it, performance-wise, to current-best nonadaptive method that can identify faulty interactions involving up to three items. We also give the first adaptive ELA-building algorithm that can identify all faulty interactions involving up to three items when safe values are known. Both of our new algorithms are generalizations of ones previously given by Mart´ınez et al. for identifying all faulty interactions involving up to two items.

Combinatorial Group Testing (CGT) is a process of identifying faulty interactions (“errors”) within a particular set of items. Error Locating Arrays (ELAs) are combinatorial designs that can be built from Covering Arrays (CAs) to not only cover all errors in a system (each involving up to a certain number of items), but to locate and identify the errors as well. In this thesis, we survey known results for CGT, as well as CAs, ELAs, and some other types of related arrays. More importantly, we give several new results. First, we give a new algorithm that can be used to test a system in which each component (factor) has two options (values), and at most two errors are present. We show that, for systems with at most two errors, our algorithm improves upon a related algorithm by Mart´ınez et al. in terms of both robustness and efficiency. Second, we give the first adaptive CGT algorithm that can identify, among a given set of k items, all faulty interactions involving up to three items. We then compare it, performance-wise, to current-best nonadaptive method that can identify faulty interactions involving up to three items. We also give the first adaptive ELA-building algorithm that can identify all faulty interactions involving up to three items when safe values are known. Both of our new algorithms are generalizations of ones previously given by Mart´ınez et al. for identifying all faulty interactions involving up to two items.

Combinatorial Group Testing (CGT) is a process of identifying faulty interactions (“errors”) within a particular set of items. Error Locating Arrays (ELAs) are combinatorial designs that can be built from Covering Arrays (CAs) to not only cover all errors in a system (each involving up to a certain number of items), but to locate and identify the errors as well. In this thesis, we survey known results for CGT, as well as CAs, ELAs, and some other types of related arrays. More importantly, we give several new results. First, we give a new algorithm that can be used to test a system in which each component (factor) has two options (values), and at most two errors are present. We show that, for systems with at most two errors, our algorithm improves upon a related algorithm by Mart´ınez et al. in terms of both robustness and efficiency. Second, we give the first adaptive CGT algorithm that can identify, among a given set of k items, all faulty interactions involving up to three items. We then compare it, performance-wise, to current-best nonadaptive method that can identify faulty interactions involving up to three items. We also give the first adaptive ELA-building algorithm that can identify all faulty interactions involving up to three items when safe values are known. Both of our new algorithms are generalizations of ones previously given by Mart´ınez et al. for identifying all faulty interactions involving up to two items.