Almost disjunctive list-decoding codes

We say that an s -subset of codewords of a binary code X is s L -bad in X if there exists an L -subset of other codewords in X whose disjunctive sum is covered by the disjunctive sum of the given s codewords. Otherwise, this s -subset of codewords is said to be s L -good in X . A binary code X is said to be a list-decoding disjunctive code of strength s and list size L (an s L -LD code) if it does not contain s L -bad subsets of codewords. We consider a probabilistic generalization of s L -LD codes; namely, we say that a code X is an almost disjunctive s L - LD code if the fraction of s L -good subsets of codewords in X is close to 1. Using the random coding method on the ensemble of binary constant-weight codes, we establish lower bounds on the capacity and error exponent of almost disjunctive s L -LD codes. For this ensemble, the obtained lower bounds are tight and show that the capacity of almost disjunctive s L -LD codes is greater than the zero-error capacity of disjunctive s L -LD codes.