  A binary code is said to be a disjunctive list-decoding $s_L$-code (LD
$s_L$-code), $s \ge 2$, $L \ge 1$, if the code is identified by the incidence
matrix of a family of finite sets in which the union (or disjunctive sum) of
any $s$ sets can cover not more than $L-1$ other sets of the family. In this
paper, we consider a similar class of binary codes which are based on a {\em
symmetric disjunctive sum} (SDS) of binary symbols. By definition, the
symmetric disjunctive sum (SDS) takes values from the ternary alphabet $\{0, 1,
*\}$, where the symbol~$*$ denotes "erasure". Namely: SDS is equal to $0$ ($1$)
if all its binary symbols are equal to $0$ ($1$), otherwise SDS is equal
to~$*$. List decoding codes for symmetric disjunctive sum are said to be {\em
symmetric disjunctive list-decoding $s_L$-codes} (SLD $s_L$-codes). In the
given paper, we remind some applications of SLD $s_L$-codes which motivate the
concept of symmetric disjunctive sum. We refine the known relations between
parameters of LD $s_L$-codes and SLD $s_L$-codes. For the ensemble of binary
constant-weight codes we develop a random coding method to obtain lower bounds
on the rate of these codes. Our lower bounds improve the known random coding
bounds obtained up to now using the ensemble with independent symbols of
codewords.
