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Self-dual codes over a family of local rings
Applicable Algebra in Engineering, Communication and Computing ( IF 0.6 ) Pub Date : 2020-10-07 , DOI: 10.1007/s00200-020-00466-4
Steven T. Dougherty , Cristina Fernández-Córdoba , Roger Ten-Valls

We construct an infinite family of commutative rings $${R_{q,\varDelta }}$$ and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from $${R_{q,\varDelta }}$$ to vectors over the residue finite field of q elements and use it to relate codes over $${R_{q,\varDelta }}$$ to codes over the finite field $${\mathbb {F}}_q$$ . Finally, we determine the parameters for when self-dual codes exist and give various constructions for self-dual codes over $${R_{q,\varDelta }}$$ .

中文翻译:

本地环族上的自对偶码

我们构建了一个无限的可交换环族 $${R_{q,\varDelta }}$$,我们研究了这些环上的代码以及这些环的结构。我们定义了一个从 $${R_{q,\varDelta }}$$ 到 q 个元素的剩余有限域上的向量的规范格雷映射,并用它来关联 $${R_{q,\varDelta }}$$ 上的代码对有限域 $${\mathbb {F}}_q$$ 进行编码。最后,我们确定存在自对偶代码时的参数,并在 $${R_{q,\varDelta }}$$ 上给出自对偶代码的各种构造。
更新日期:2020-10-07
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