论文标题
两点代数几何代码的标志中的等轴测二重属性
The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes
论文作者
论文摘要
代码的标志$ C_0 \ subsetNeq C_1 \ subsetNeq \ cdots \ subsetNeq c_s \ subseteq {\ mathbb f} _q^n $可以满足{\ is ismetry-dual property}的满足，如果存在$ {\ bf x} \ n $ n.代码$ c_i $ is {\ bf x} - 对偶代码$ c_ {s-i}^\ perp $的iSometric，用于所有$ i = 0，\ ldots，s $。对于$ p $和$ q $合理的位置$ {\ MATHCAL F} $，我们调查了两点代数几何代码代码$ c_ \ mathcal l（d，d，d，a_0p+bq+bq c _ sebset c_ \ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq c_ bq） \ subsetNeq c_ \ Mathcal l（d，a_sp+bq），$$，其中除数$ d $是$ {\ mathcal f} $的成对不同理性位置和$ p，$ p，q $不在$ \ mbox {supp}（supp}（d）$中的总和。我们以通用功能字段的$ b $来表征这些序列。然后，我们将结果应用于广泛的Kummer Extensions $ {\ MATHCAL F} $由affine方程式定义的$ y^m = f（x）$，对于$ f（x）$ a $ r $的可分开多项式$ r $，其中$ \ mbox {gcd}（gcd}（r，m）= 1 $。对于$ p $，无限的理性地方和$ q $与$ f（x）$的根之一相关的理性地方，这表明两点代数几何代码的旗帜在且仅当$ m $ dif $ 2b+1 $时，具有等轴测件属性。最后，我们通过将它们应用于几个良好的功能字段上的两点代码来说明结果。
A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field ${\mathcal F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of ${\mathcal F}$ and $P, Q$ are not in $\mbox{supp}(D)$. We characterize those sequences in terms of $b$ for general function fields. We then apply the result to the broad class of Kummer extensions ${\mathcal F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.
