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.

翻译：设$a$和$b$是有限域$\mathbb{F}_q$（其中$q>2$）中的两个非零元。已有结果表明，若$a$与$b$在$\mathbb{F}_q$中具有相同的乘法阶，则$\mathbb{F}_q$上的$a$-常循环码与$b$-常循环码族是单项等价的。本文研究当$a$和$b$具有不同乘法阶时，$a$-常循环码与$b$-常循环码之间的单项等价性。我们提出了建立此类常循环码单项等价的新条件，超越了以往判定单项等价常循环码与循环码的方法。作为应用，我们利用这些结果更系统地搜索新的线性码。特别地，我们给出了超过70个在不同有限域上的新纪录线性码，以及新的二进制量子码。