Anthony J. Macula

A simple construction of d -disjunct matrices with certain constant weights

Abstract We give a simple method of constructing d -disjunct matrices. For k > d , our construction yields a ( d n ) × ( k n ) d -disjunct matrix with column weight ( d k ) and row weight ( k − d n − d ).

New constructions of superimposed codes

Kautz-Singleton (1964) suggested a class of binary superimposed codes which are based on the q-ary Reed-Solomon codes (RS codes). Applying a concatenation of the binary constant-weight error-correcting codes and the shortened RS codes, we obtain new constructions of superimposed codes. Tables of their parameters are given. From the tables it follows that the rate of obtained codes exceeds the corresponding random coding bound.

Families of Finite Sets in which No Intersection of ℓ Sets Is Covered by the Union of s Others

Abstract In 1964, Kautz and Singleton ( IEEE Trans. Inform. Theory 10 (1964), 363–377) introduced the superimposed code concept. A binary superimposed code of strength s is identified by the incidence matrix of a family of finite sets in which no set is covered by the union of s others ( J. Combin. Theory Ser. A 33 (1982), 158–166 and Israel J. Math. 51 (1985), 75–89). In the present paper, we consider a generalization called a binary superimposed ( s,l )-code which is identified by the incidence matrix of a family defined in the title. We discuss the constructions based on MDS-codes (The Theory of Error-correcting Codes, North-Holland, Amsterdam, The Netherlands, 1983) and derive upper and lower bounds on the rate of these codes.

Error-correcting nonadaptive group testing with d e -disjunct matrices

Abstract d-disjunct matrices constitute a basis for nonadaptive group testing (NGT) algorithms and binary d-superimposed codes. The rows of a d-disjunct matrix represent the tests in a NGT algorithm which identifies up to d defects in a population. The columns of a d-disjunct matrix represent binary d-superimposable codewords. A d-disjunct matrix μ is called de-disjunct if given any d + 1 columns of μ with one designated, there are e + 1 rows with a 1 in the designated column and a 0 in each of the other d columns. de-disjunct matrices form a basis for e error-correcting NGT algorithms. In this paper, we construct de-disjunct matrices. In so doing, we simultaneously construct e error-correcting binary d-superimposed codes. The results of this paper can be used to construct pooling designs for the screening recombinant DNA libraries. Such screenings are a major component of the Human Genome Project.

Probabilistic nonadaptive group testing in the presence of errors and DNA library screening

We use the subset containment relation to construct a probabilistic nonadaptive group testing design and decoding algorithm that, in the presence of testing errors, identifies many positives in a population. We give a lower bound for the expected portion of positives identified as a function of an upper bound on the number of testing errors.

Exordium for DNA Codes

We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNA-sequence codewords. This enhancement involves abstractions of the pertinent aspects of DNA; it nevertheless ensures specificity of binding for all pairs of single strands derived from its codewords—the key desideratum of DNA codes– i.e. with binding feasible only between reverse complementary strands. We defer discussing the combinatorial-optimization superincumbencies of code construction. Generalization of deletion similarity to an optimal sequence-alignment score could readily effect advantageous improvements (Kaderali, Masters Thesis, Informatics, U. Köln, 2001) but would render the combinatorics opaque. We mention motivating applications of DNA codes.

Probabilistic Nonadaptive and Two-Stage Group Testing with Relatively Small Pools and DNA Library Screening

We use a simple, but nonstandard, incidence relation to construct probabilistic nonadaptive group testing algorithms that identify many positives in a population with a zero probability of yielding a false positive. More importantly, we give two-stage modifications of our nonadaptive algorithms that dramatically reduce the expected number of sufficient pools. For our algorithms, we give a lower bound on the probability of identifying all the positives and we compute the expected number of positives identified. Our method gives control over the pool sizes. In DNA library screening algorithms, where relatively small pools are generally preferred, having control over the pool sizes is an important practical consideration.

New Applications and Results of Superimposed Code Theory Arising from the Potentialities of Molecular Biology

Superimposed codes (SC) were introduced by Kautz-Singleton (1964) [1], who worked out the important constructive methods. Dyachkov-Rykov [2, 3, 4, 5, 6] and Erdos-Frankl-Furedi [7] obtained upper and lower bounds on the rate of SC. Dyachkov-Macula-Rykov [8, 9, 10, 11] investigated the development of constructions for SC (nonadaptive pooling designs) intended for the clone-library screening problem. (See Balding-Torney [12] and Knill-Bruno-Torney [13]). In this paper, we give an introduction to the problem and a detailed survey of our recent results on constructive methods of SC. We discuss superimposed distance codes and list-decoding superimposed codes.

Trivial two-stage group testing for complexes using almost disjunct matrices

Abstract Let [t] represent a finite population with t elements. Suppose we have an unknown d-family of k-subsets Γ of [t]. We refer to Γ as the set of positive k-complexes. In the group testing for complexes problem, Γ must be identified by performing 0, 1 tests on subsets or pools of [t]. A pool is said to be positive if it completely contains a complex; otherwise the pool is said to be negative. In classical group testing, each member of Γ is a singleton. In this paper, we exhibit and analyze a probabilistic trivial two-stage algorithm that identifies the positive complexes.

Two Models of Nonadaptive Group Testing for Designing Screening Experiments

We discuss two non-standard models of nonadaptive combinatorial search which develop the conventional disjunct search model of Du and Hwang (1993) for a small number of defective elements contained in a finite ground set or a population. The first model called asearch of defective supersets (complexes)was suggested in D’yachkovetal. (2000c,d). The second model which can be called asearch of defective subsets in the presence of inhibitorswas introduced for the case of an adaptive search by Farachetal. (1997) and De Bonis and Vaccaro (1998). For these models, we study the constructive search methods based on the known constructions for the disjunct model from Kautz and Singleton (1964) and from D’yachkovetal. (2000a,b).