No 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 ðs; 'Þ-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. # 2002 Elsevier Science (USA)

The screening of data sets for "positive data objects" is essential to modern technology. A (group) test that indicates whether a positive data object is in a specific subset or pool of the dataset can greatly facilitate the identification of all the positive data objects. A collection of tested pools is called a pooling design. Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we use the (linear) subspace relation coupled with the general concept of a "containment matrix" to construct pooling designs with surprisingly high degrees of error correction (detection.) Error-correcting pooling designs are important to biotechnical applications where error rates often are as high as 15%. What is also surprising is that the rank of the pooling design containment matrix is independent of the number of positive data objects in the dataset.

No abstract

DNA nanotechnology often requires collections of oligonucleotides called "DNA free energy gap codes" that do not produce erroneous crosshybridizations in a competitive muliplexing environment. This paper addresses the question of how to design these codes to accomplish a desired amount of work within an acceptable error rate. Using a statistical thermodynamic and probabilistic model of DNA code fidelity and mathematical random coding theory methods, theoretical lower bounds on the size of DNA codes are given. More importantly, DNA code design parameters (e.g., strand number, strand length and sequence composition) needed to achieve experimental goals are identified.