Bicyclic codes are a generalization of the one-dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length \(n^2\) contains its dual if and only if its design distance is lower than \(n-O(\sqrt{n})\). We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length \(n^2\) contains Hermitian dual if and only if its design distance is lower than \(n-O(\sqrt{n})\). Our results are analogous to some structural results known for BCH and Reed–Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.

中文翻译：

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量子双环双曲码

双循环码是将一维（1D）循环码概括为二维（2D）的概括。与1D情况类似，在某些情况下，也可以构造2D循环码以保证指定的最小距离。这些代码的许多方面尚未开发。由于构造量子码的问题，我们研究了某些双环码的一些结构性质。我们证明了，当且仅当其设计距离小于\（nO（\ sqrt {n}）\）时，长度为\（n ^ 2 \）的原始狭义双环双曲代码才包含其对偶。我们也将充分条件扩展到非原始情况。我们还表明，在二次扩展字段上，长度为（（n ^ 2 \））的原始双环双曲代码当且仅当其设计距离小于\（nO（\ sqrt {n}）\）时，包含Hermitian对偶。我们的结果类似于BCH和Reed-Solomon码的一些结构性结果。它们加深了我们对双环编码的理解。我们还通过显示我们可以根据我们的结果构造两类量子双环代码来应用这些结果。