论文标题
多扭码代码作为主要理想域上的免费模块
Multi-twisted codes as free modules over principal ideal domains
论文作者
论文摘要
我们通过在有限字段上引入循环代码的简单代数结构来开始本章。这种结构经历了一系列概括，以呈现constacyclic，Quasi-cyclic（QC），准串联（QT），概括的准循环（GQC）和多键（MT）代码的代数描述。这些代码与免费$ \ mathbb {f} _q [x] $ - module $ \ left（\ mathbb {f} _q [x] \ right）^\ ell $之间的对应关系。因此，这些代码中的任何一个都对应于主体理想域（PID）$ \ mathbb {f} _Q [x] $上的免费线性代码。该代码的基础是存在，用于构建具有多项式条目的发电机矩阵，称为发电机多项式矩阵（GPM）。将PID上的矩阵的赫米特法线形式开发，以实现MT代码的降低的GPM。引入了降低的GPM的某些特性，例如，相同的方程式。给出了MT代码的双代码$ \ Mathcal {C}^\ PERP $的GPM公式。此时，特别注意QC代码。对于QC代码$ \ MATHCAL {C} $，我们定义其反向代码$ \ Mathcal {R} $。如果$ \ Mathcal {r} = \ Mathcal {C} $或$ \ Mathcal {C}^\ Perp = \ Mathcal {C} $，我们将调用$ \ MATHCAL {C} $可逆或自动划分。给出了$ \ Mathcal {r} $的GPM的公式。我们表征了QC代码的GPM，这些QC代码结合了可逆性和自我二重性/自治性。对于有兴趣运行计算机搜索最佳代码的读者，我们显示了具有最著名的参数为线性代码的二进制自动可逆QC代码的存在。这些结果可以通过使用符合上述特征的GPM来通过蛮力搜索来获得。
We begin this chapter by introducing the simple algebraic structure of cyclic codes over finite fields. This structure undergoes a series of generalizations to present algebraic descriptions of constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes. The correspondence between these codes and submodules of the free $\mathbb{F}_q[x]$-module $\left(\mathbb{F}_q[x]\right)^\ell$ is established. Thus, any of these codes corresponds to a free linear code over the principal ideal domain (PID) $\mathbb{F}_q[x]$. A basis of this code exists and is used to build a generator matrix with polynomial entries, called the generator polynomial matrix (GPM). The Hermite normal form of matrices over PIDs is exploited to achieve the reduced GPMs of MT codes. Some properties of the reduced GPM are introduced, for example, the identical equation. A formula for a GPM of the dual code $\mathcal{C}^\perp$ of a MT code is given. At this point, special attention is paid to QC codes. For a QC code $\mathcal{C}$, we define its reversed code $\mathcal{R}$. We call $\mathcal{C}$ reversible or self-dual if $\mathcal{R}=\mathcal{C}$ or $\mathcal{C}^\perp=\mathcal{C}$, respectively. A formula for a GPM of $\mathcal{R}$ is given. We characterize GPMs for QC codes that combine reversibility and self-duality/self-orthogonality. For the reader interested in running computer search for optimal codes, we show the existence of binary self-orthogonal reversible QC codes that have the best known parameters as linear codes. These results can be obtained by brute-force search using GPMs that meet the above characterization.
