Isometry-Dual Flags of Many-Point AG Codes

Let $Fq$ be a finite field. A flag of $Fq$-linear codes $C0\subsetneq C1\subsetneq\dots\subsetneq Cs$ is said to satisfy the isometry-dual property if there exists a vector $x\in(Fq^*)n$ such that $Ci=x\cdot C{s-i}^\perp$, where $Ci^\perp$ denotes the dual code of $Ci$. Consider $F/Fq$ a function field and let $P$ and $Q1,\ldots,Qt$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q1,\dots,Qt$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a0P+bQ1)\subsetneq C(D,a1P+bQ1))\subsetneq\dots\subsetneq C(D,asP+bQ1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a0,\dots,as$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a0P+\sum{i=1}^t\betaiQi)\subsetneq C(D, a1P+\sum{i=1}^t\betaiQi))\subsetneq\dots\subsetneq C(D, asP+\sum{i=1}^t\betaiQi)$ for any tuple of (either positive or negative) integers $\beta1,\dots,\betat$ and for an increasing sequence of (either positive or negative) integers $a0,\dots,as$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(Fq(x, y)/Fq(x))$-invariant sum of rational places of $F$ such that $P,Qi\notin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $\beta_i$'s such that the flag has the isometry-dual property.