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首页> 外文期刊>IEEE Transactions on Information Theory >Hermitian Self-Dual Abelian Codes
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Hermitian Self-Dual Abelian Codes

机译：厄米（Hermitian）自对偶阿贝尔密码

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摘要

Hermitian self-dual abelian codes in a group ring $BBF_{q^{2}}[G]$, where $BBF_{q^{2}}$ is a finite field of order $q^{2}$ and $G$ is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in $BBF_{q^{2}}[G]$ is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in $BBF_{q^{2}}[G]$, i.e., there exists a Hermitian self-dual abelian code in $BBF_{q^{2}}[G]$ if and only if the order of $G$ is even and $q=2^{l}$ for some positive integer $l$. Later on, the study is further restricted to the case where $BBF_{2^{2l}}[G]$ is a principal ideal group ring, or equivalently, $Gcong AoplusBBZ_{2^{k}}$ with $2nmidvert Avert$ . Based on the characterization obtained, the number of Hermitian self-dual abelian codes in $BBF_{{2}^{2l}}[AoplusBBZ_{2^{k}}]$ can be determined easily. When $A$ is cyclic, th- s result answers an open problem of Jia concerning Hermitian self-dual cyclic codes. In many cases, $BBF_{{2}^{2l}}[AoplusBBZ_{2^{k}}]$ contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of $l$ and the order of $A$. Finally, the distribution of finite abelian groups $A$ such that a unique Hermitian self-dual abelian code exists in $BBF_{{2}^{2l}}[AoplusBBZ_{2}]$ is established, together with the distribution of odd integers $m$ such that a unique Hermitian self-dual cyclic code of length 2 m over $BBF_{{2}^{2l}}$ exists.
机译：组环$ BBF_ {q ^ {2}} [G] $中的埃尔米特自对偶阿贝尔编码，其中$ BBF_ {q ^ {2}} $是阶数$ q ^ {2} $和$的有限域G $是一个有限的阿贝尔群，正在研究中。使用众所周知的半单群环离散傅里叶变换分解，给出了$ BBF_ {q ^ {2}} [G] $中的埃尔米特自对偶阿贝尔编码的特征，以及必要和充分的替代证明。 $ BBF_ {q ^ {2}} [G] $中存在这样的代码的条件，即，如果和，则$ BBF_ {q ^ {2}} [G] $中存在一个埃尔米特自对偶阿贝尔代码仅当$ G $的阶数为偶数且$ q = 2 ^ {l} $对于某个正整数$ l $时。后来，该研究进一步局限于$ BBF_ {2 ^ {2l}} [G] $是主要理想群环的情况，或者等效为$ Gcong AoplusBBZ_ {2 ^ {k}} $与$ 2nmidvert Avert的情况。 $。根据获得的特征，可以轻松确定$ BBF _ {{{2} ^ {2l}} [AoplusBBZ_ {2 ^ {k}}] $中的埃尔米特自对偶阿贝尔编码的数量。当$ A $是循环的时，此结果回答了关于Hermitian自对偶循环码的Jia的开放问题。在许多情况下，$ BBF _ {{2} ^ {2l}} [AoplusBBZ_ {2 ^ {k}}] $包含唯一的Hermitian自对偶阿贝尔代码。这种情况的标准是根据$ l $和$ A $的顺序确定的。最后，建立有限的阿贝尔群$ A $的分布，以便在$ BBF _ {{{2} ^ {2l}} [AoplusBBZ_ {2}] $中存在唯一的埃尔米特自对偶阿贝尔编码，并建立奇数分布整数$ m $，使得存在$ BBF _ {{{2} ^ {2l}} $$上长度为2 m的唯一Hermitian自对偶循环码。

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