Almost cover-free codes and designs

An s-subset of codewords of a binary code X is said to be $$(s,\\,\\ell )$$(s,ℓ)-bad in X if the code X contains a subset of $$\\ell $$ℓ other codewords such that the conjunction of the $$\\ell $$ℓ codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called $$(s,\\,\\ell )$$(s,ℓ)-good in X. A binary code X is said to be a cover-free (CF) $$(s,\\,\\ell )$$(s,ℓ)-code if the code X does not contain $$(s,\\,\\ell )$$(s,ℓ)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF $$(s,\\,\\ell )$$(s,ℓ)-codes, namely: a binary code X is said to be an almost CF $$(s,\\,\\ell )$$(s,ℓ)-code if the relative number of its $$(s,\\,\\ell )$$(s,ℓ)-good s-subsets is close to 1. We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes. Our main result shows that the capacity for almost CF $$(s,\\,\\ell )$$(s,ℓ)-codes is essentially greater than the rate for ordinary CF $$(s,\\,\\ell )$$(s,ℓ)-codes.