  This paper studies the cardinality of codes correcting insertions and
deletions. We give improved upper and lower bounds on code size. Our upper
bound is obtained by utilizing the asymmetric property of list decoding for
insertions and deletions and can be seen as analogous to the Elias bound in the
Hamming metric. Our non-asymptotic bound is better than the existing bounds
when the minimum Levenshtein distance is relatively large. The asymptotic bound
exceeds the Elias and the MRRW bounds adapted from the Hamming-metric bounds
for the binary and the quaternary cases. Our lower bound improves on the bound
by Levenshtein, but its effect is limited and vanishes asymptotically.
