  Cyclic codes are among the most important families of codes in coding theory
for both theoretical and practical reasons. Despite their prominence and
intensive research on cyclic codes for over a half century, there are still
open problems related to cyclic codes. In this work, we use recent results on
the equivalence of cyclic codes to create a more efficient algorithm to
partition cyclic codes by equivalence based on cyclotomic cosets. This
algorithm is then implemented to carry out computer searches for both cyclic
codes and quasi-cyclic (QC) codes with good parameters. We also generalize
these results to repeated-root cases. We have found several new linear codes
that are cyclic or QC as an application of the new approach, as well as more
desirable constructions for linear codes with best known parameters. With the
additional new codes obtained through standard constructions, we have found a
total of 14 new linear codes.
