  We introduce new sufficient conditions for permutation and monomial
equivalence of linear cyclic codes over various finite fields. We recall that
monomial equivalence and isometric equivalence are the same relation for linear
codes over finite fields. A necessary and sufficient condition for the monomial
equivalence of linear cyclic codes through a shift map on their defining set is
also given. Moreover, we provide new algebraic criteria for the monomial
equivalence of constacyclic codes over $\mathbb{F}_4$. Finally, we prove that
if $\gcd(3n,\phi(3n))=1$, then all permutation equivalent constacyclic codes of
length $n$ over $\mathbb{F}_4$ are given by the action of multipliers. The
results of this work allow us to prune the search algorithm for new linear
codes and discover record-breaking linear and quantum codes.
