  Linear computation broadcast (LCBC) refers to a setting with $d$ dimensional
data stored at a central server, where $K$ users, each with some prior linear
side-information, wish to retrieve various linear combinations of the data. The
goal is to determine the minimum amount of information that must be broadcast
to satisfy all the users. The reciprocal of the optimal broadcast cost is the
capacity of LCBC. The capacity is known for up to $K=3$ users. Since LCBC
includes index coding as a special case, large $K$ settings of LCBC are at
least as hard as the index coding problem. Instead of the general setting (all
instances), by focusing on the generic setting (almost all instances) this work
shows that the generic capacity of the symmetric LCBC (where every user has
$m'$ dimensions of side-information and $m$ dimensions of demand) for large
number of users ($K>d$ suffices) is $C_g=1/\Delta_g$, where
$\Delta_g=\min\left\{\max\{0,d-m'\}, \frac{dm}{m+m'}\right\}$, is the broadcast
cost that is both achievable and unbeatable asymptotically almost surely for
large $n$, among all LCBC instances with the given parameters $p,K,d,m,m'$.
Relative to baseline schemes of random coding or separate transmissions, $C_g$
shows an extremal gain by a factor of $K$ as a function of number of users, and
by a factor of $\approx d/4$ as a function of data dimensions, when optimized
over remaining parameters. For arbitrary number of users, the generic capacity
of the symmetric LCBC is characterized within a factor of $2$.
