码的自正交旗与代数几何码旗的平移
Self-orthogonal flags of codes and translation of flags of algebraic geometry codes
AI总结：
本文研究线性码旗的自正交性与等距对偶性质，刻画完全等距对偶旗的条件，证明代数几何码旗的平移性质并构造完全自正交旗，最后刻画相关除子与有限多个等距向量的循环关系。
摘要：
线性码的一个旗 $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n$ 被称为自正交的，如果该旗中各码的对偶满足 $C_{i}^\perp=C_{s-i}$；并且称其关于等距向量 ${\bf x}$ 满足等距对偶性质，如果对 $i=1, \dots, s$ 有 $C_i^\perp={\bf x} C_{s-i}$。我们借助 ${\mathbb F}_q^n$ 的某个线性子空间中具有非零坐标的字的存在性，刻画了具有等距对偶性质的完全（即 $s=n$）旗。对于代数几何（AG）码的旗，我们证明了等距对偶旗的一个所谓平移性质，并给出了完全自正交旗的一种构造，在某些极大函数域上提供了自正交旗的例子。最后，我们刻画了给出等距对偶性质的除子以及相关的等距向量，表明对每个函数域只存在有限多个等距向量，并且它们通过循环重复相互关联。
英文摘要：
A flag $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n $ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_{i}^\perp=C_{s-i}$, and it is said to satisfy the isometry-dual property with respect to an isometry vector ${\bf x}$ if $C_i^\perp={\bf x} C_{s-i}$ for $i=1, \dots, s$. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of ${\mathbb F}_q^n$. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
