论文标题
将线性代码嵌入自动代码及其最佳最小距离中
Embedding linear codes into self-orthogonal codes and their optimal minimum distances
论文作者
论文摘要
我们根据其发电机矩阵中的列数量的数量来获得给定二进制线性代码的自动性的表征，该列矩阵扩展了Bouyukliev等人的结果。 （2006）。作为一个应用程序，我们提供了一种算法方法，将给定的二进制$ k $二维线性代码$ \ MATHCAL {C} $（$ k = 2,3,4 $）嵌入最短的代码中，该代码具有最短的长度，该代码具有相同的尺寸$ k $ and v $ k $ and ve'\ ge d'\ ge d'\ ge d'\ ge d'\ ge d'\ ge Mathcalcal {c} $。对于$ k> 4 $，我们建议一种递归方法将$ k $维线性代码嵌入自动代码中。我们还为任何长度$ n $的最低距离的最小距离提供了新的明确公式，任何长度为4 $和任何长度$ n \ not \ equiv 6,13,14,21,21,21,22,22,22,22,22,22,22,22,28,29 \ pmod {31} $ with dimension with dimension code copt $ n dimensive fime $ n of $ n $ n $ n，$ n \ equiv 5的最佳最佳均可确定$ 5。 li-xu-zhao（2008）当$ n \ equiv 0,3,4,5,10,11,12 \ pmod {15} $。然后，使用岩浆，我们观察到我们的嵌入将最佳的线性代码发送到最佳的自动代码。
We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $\mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n \not\equiv 6,13,14,21,22,28,29 \pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n \equiv 0,3,4,5,10,11,12 \pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code.
