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.

翻译：我们为线性循环编码的异位和单一等同提出了新的充分条件,用于不同限定领域的线性循环编码的变位和单一等值。我们回顾,单分子等同和等同是线性编码与有限领域的线性编码的相同关系。我们还提供了一个必要和充分的条件,通过对线性循环编码定义组的变换图,使线性循环编码的单分子等同具有必要和充分的条件。此外,我们还为共性编码的单分子等同提供了新的代数标准,超过$\mathbb{F ⁇ 4美元。最后,我们证明,如果$\gcd(3n,\phi(3n))=1美元,那么所有长度超过$\mathbb{F ⁇ 4美元的等等同的线性等值编码都是由乘数动作给出的。这项工作的结果使我们得以对新的线性编码的搜索算法进行精细化,并发现破记录的线性和量性线性编码。