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.