  In the first part of the paper, we consider the relation between kissing
number and the secrecy gain. We show that on an $n=24m+8k$-dimensional even
unimodular lattice, if the shortest vector length is $\geq 2m$, then as the
number of vectors of length $2m$ decreases, the secrecy gain increases. We will
also prove a similar result on general unimodular lattices. We will also
consider the situations with shorter vectors. Furthermore, assuming the
conjecture by Belfiore and Sol\'e, we will calculate the difference between
inverses of secrecy gains as the number of vectors varies. We will show by an
example that there exist two lattices in the same dimension with the same
shortest vector length and the same kissing number, but different secrecy
gains. Finally, we consider some cases of a question by Elkies by providing an
answer for a special class of lattices assuming the conjecture of Belfiore and
Sol\'e. We will also get a conditional improvement on some Gaulter's results
concerning the conjecture.
