Let a and b be two non-zero elements of a finite field \(\mathbb {F}_q\), where \(q>2\). It has been shown that if a and b have the same multiplicative order in \(\mathbb {F}_q\), then the families of a-constacyclic and b-constacyclic codes over \(\mathbb {F}_q\) are monomially equivalent. In this paper, we investigate the monomial equivalence of a-constacyclic and b-constacyclic codes when a and b have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than 70 new record-breaking linear codes over various finite fields, as well as new binary quantum codes.

设 a 和 b 是有限域 \（\mathbb {F}_q\） 的两个非零元素，其中 \（q>2\）。已经表明，如果 a 和 b 在 \（\mathbb {F}_q\） 中具有相同的乘法顺序，那么 \（\mathbb {F}_q\） 上的 a 常环和 b 常环码族在单项式上是等价的。在本文中，我们研究了当 a 和 b 具有不同的乘法顺序时 a-constacyclic 和 b-constacyclic 代码的单项式等价性。我们提出了在此类恒环代码中建立单项式等价的新条件，超越了以前确定单项式等价常循环和循环代码的方法。作为应用程序，我们使用这些结果来更系统地搜索新的线性代码。特别是，我们提出了 70 多个在各种有限域上打破纪录的新线性代码，以及新的二进制量子代码。