Let ${\mathbb{F}}_q$ be a finite field and G a finite abelian group. An abelian code is an ideal of ${\mathbb{F}}_q[G]$. Two abelian codes ${\mathfrak{c}}_1$ and ${\mathfrak{c}}_2$ of ${\mathbb{F}}_q[G]$ are equivalent if there exists an automorphism of G whose linear extension to ${\mathbb{F}}_q[G]$ maps ${\mathfrak{c}}_1$ onto ${\mathfrak{c}}_2$. MacWilliams determined the number of equivalent classes of minimal abelian codes (minimal ideals) in ${\mathbb{F}}_2[G]$ for cyclic group G of odd cardinality. Miller claimed that MacWilliams' result remains true in general, which is however not correct as pointed out by Ferraz, Guerreiro and Polcino Milies. In this paper we completely determine the number of equivalent classes of minimal abelian codes for any ${\mathbb{F}}_q[G]$.

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关于最小阿贝尔码的等价类

让${\mathbb{F}}_q$是一个有限域，G是一个有限阿贝尔群。阿贝尔码是一个理想的${\mathbb{F}}_q[G]$. 两个阿贝尔码${\mathfrak{C}}_1$和${\mathfrak{C}}_2$的${\mathbb{F}}_q[G]$如果存在G的自同构，其线性扩展为${\mathbb{F}}_q[G]$地图${\mathfrak{C}}_1$到${\mathfrak{C}}_2$. MacWilliams 确定了在${\mathbb{F}}_2[G]$对于奇数基数的循环群G。米勒声称麦克威廉姆斯的结果总体上仍然正确，但正如费拉兹、格雷罗和波尔奇诺米利斯所指出的那样，这并不正确。在本文中，我们完全确定了任何最小阿贝尔码的等效类数${\mathbb{F}}_q[G]$.