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Some new results on superimposed codes. (English) Zbl 1169.94350
Summary: A $$(w,r)$$ cover-free family is a family of subsets of a finite set such that no intersection of $$w$$ members of the family is covered by a union of $$r$$ others. A $$(w,r)$$ superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as the concept of key distribution pattern. In the present paper, we give some new results on superimposed codes. First we construct superimposed codes from super-simple designs which give us results better than superimposed codes constructed by other known methods. Next we prove the uniqueness of the (1,2) superimposed code of size $$9 \times 12$$, the (2,2) superimposed code of size $$14 \times 8$$, and the (2,3) superimposed code of size $$30 \times 10$$. Finally, we improve numerical values of upper bounds for the asymptotic rate of some ($$w,r$$) superimposed codes.

##### MSC:
 94B25 Combinatorial codes 05B05 Combinatorial aspects of block designs
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