Let \(\theta \) be an automorphism on a finite field \(\mathbb {F}_q.\) In this paper, we give a way to construct and enumerate Euclidean self-dual \(\theta \)-cyclic codes of length n over \(\mathbb {F}_q\) when n is even and \(\gcd (n,|\theta |)=1.\) The restriction \(\gcd (n,|\theta |)=1\) implies that the \(\theta \)-cyclic codes are in fact cyclic codes and \(q=2^m,\) for some integer \(m\ge 1.\) The construction and enumeration are done by analyzing the orbits of cyclotomic cosets under a multiplier map induced by \(\theta .\)

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关于欧几里得自对偶斜循环码的构造和枚举的说明

令\（\ theta \）是有限域\（\ mathbb {F} _q。\）的自同构。在本文中，我们提供了一种构造和枚举欧几里德自对偶\（\ theta \）循环码的方法当n为偶数且\（\ gcd（n，| \ theta |）= 1。\\）时长度n超过\（\ mathbb {F} _q \）的限制\（\ gcd（n，| \ theta |） = 1 \）表示\（\ theta \） -循环码实际上是循环码，对于某些整数\（m \ ge 1. \）是\（q = 2 ^ m，\）构造和枚举已完成通过分析由\（\ theta。\）引起的乘数映射下的环行同集的轨道