On equivalent classes of minimal Abelian codes
On equivalent classes of minimal Abelian codes
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DOI：
10.1016/j.jcta.2022.105616
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发表时间：
2022-07
期刊：
影响因子：
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通讯作者：
Yuanhong Ren;Dongchun Han
中科院分区：
文献类型：
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作者：
Yuanhong Ren;Dongchun Han
设Fq是有限域，G是有限交换群.阿贝尔码是Fq [G]的理想.若存在G的一个自同构，其对Fq [G]的线性扩张将c1映射到c2上，则Fq [G]的两个交换码c1和c2是等价的. MacWilliams确定了奇基数循环群G的极小交换码（极小理想）在F2 [G]中的等价类的个数.米勒声称，麦克威廉姆斯的结果仍然是正确的，但这是不正确的，因为指出了费拉斯，格雷罗和波尔奇诺迈尔斯。本文完全确定了任意Fq [G]的极小阿贝尔码的等价类的个数。
Let F q be a finite field and G a finite abelian group. An abelian code is an ideal of F q [G]. Two abelian codes c 1 and c 2 of F q [G] are equivalent if there exists an automorphism of G whose linear extension to F q [G] maps c 1 onto c 2. MacWilliams determined the number of equivalent classes of minimal abelian codes (minimal ideals) in 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 F q [G].