Almost Disjunctive List-Decoding Codes
  A binary code is said to be a disjunctive list-decoding $s_L$-code, $s\ge1$, $L\ge1$, (briefly, LD $s_L$-code) if the code is identified by the incidence matrix of a family of finite sets in which the union of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we introduce a natural {\em probabilistic} generalization of LD $s_L$-code when the code is said to be an almost disjunctive LD $s_L$-code if the unions of {\em almost all} $s$ sets satisfy the given condition. We develop a random coding method based on the ensemble of binary constant-weight codes to obtain lower bounds on the capacity and error probability exponent of such codes. For the considered ensemble our lower bounds are asymptotically tight. 
A.G. Dyachkov, I.V. Vorobyev, N.A. Polyanskii, V.Yu. Shchukin
cs.IT math.IT