  The $b$-symbol read channel is motivated by the limitations of the reading
process in high density data storage systems. The corresponding new metric is a
generalization of the Hamming metric known as the $b$-symbol weight metric and
has become an important object in coding theory. In this paper, the general
$b$-symbol weight enumerator formula for irreducible cyclic codes is presented
by using the Gaussian period and a new invariant $\#U(b,j,N_1)$. The related
$b$-symbol weight hierarchies $\{d_1(\C),d_2(\C),\ldots,d_K(\C)\}$
($K=\dim(\C)$) are given for some cases. The shortened codes which are optimal
from some classes of irreducible cyclic codes are given, where the shorten set
$\mathcal{T}$ is the complementary set of $b$-symbol support of some codeword
with the minimal $b$-symbol weight.
