Generator Polynomial Matrices of Reversed and Reversible Quasi-Cyclic Codes
Generator Polynomial Matrices of Reversed and Reversible Quasi-Cyclic Codes
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发表时间：
2020-10
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通讯作者：
R. T. Eldin;H. Matsui
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作者：
R. T. Eldin;H. Matsui
有限字段$ {\ Mathbb {f} _Q} $对某些$ {\ Mathbb {f} _Q} [x] $模块对应的Quasi-Cycle（QC）代码对应} $由发电机多项式矩阵。 MATHCAL {C} $，然后C是这项工作的可逆代码，我们证明了一个发电机的多项式矩阵的反向代码ℛ处方的QC代码$ \ Mathcal {C C} $。在确保任何QC代码的可逆性方面，我们都表征了可逆的QC代码的多项式材料，我们将我们的理论结果应用于具有最著名的参数的QC代码。计算机搜索用于显示可逆的QC代码，这些QC代码在线性代码的最小距离上实现了上限。
Quasi-cyclic (QC) codes over the finite field ${\mathbb{F}_q}$ correspond to certain ${\mathbb{F}_q}[x]$-modules. We describe a QC code $\mathcal{C}$ by a generator polynomial matrix. The code obtained by reversing the codewords of $\mathcal{C}$ is called the reversed code of $\mathcal{C}$ and denoted by ℛ. If$\mathcal{R} = \mathcal{C}$, then C is a reversible code. In this work, we prove a formula for a generator polynomial matrix of the reversed code ℛ of a prescribed QC code $\mathcal{C}$. Then, we present a necessary and sufficient condition in terms of the proven formula to ensure the reversibility of any QC code. Moreover, we characterize the reduced generator polynomial matrices of reversible QC codes. As an application, we apply our theoretical results to QC codes with the best known parameters. Computer search is used to show the existence of reversible QC codes that achieve the upper bounds on the minimum distance of linear codes. Several binary reversible QC codes with the best known parameters are provided.