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Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

Maria Bras-Amorós, Alonso S. Castellanos, Luciane Quoos

TL;DR

The paper studies flags of linear codes with the isometry-dual property, focusing on complete flags built from algebraic geometry codes. It develops a linear-algebraic certificate via nullspaces of parity-product matrices and a divisor-theoretic framework to characterize when flags satisfy isometry-dualism; it then shows a translation property that preserves the isometry-dual condition under suitable divisors, and provides practical constructions of self-orthogonal flags in characteristic two using interpolating functions and divisor data. The results include explicit constructions over maximal function fields (including Kummer and Hermitian cases), a complete divisors-based characterization of isometry-dual flags through canonical divisor conditions, and a finite, structured description of the associated isometry vectors. Overall, the work connects AG-code flag geometry, divisor theory, and explicit constructions to yield self-orthogonal flags with potential quantum coding applications and insights into the structure of isometry vectors across function fields.

Abstract

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.

Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

TL;DR

The paper studies flags of linear codes with the isometry-dual property, focusing on complete flags built from algebraic geometry codes. It develops a linear-algebraic certificate via nullspaces of parity-product matrices and a divisor-theoretic framework to characterize when flags satisfy isometry-dualism; it then shows a translation property that preserves the isometry-dual condition under suitable divisors, and provides practical constructions of self-orthogonal flags in characteristic two using interpolating functions and divisor data. The results include explicit constructions over maximal function fields (including Kummer and Hermitian cases), a complete divisors-based characterization of isometry-dual flags through canonical divisor conditions, and a finite, structured description of the associated isometry vectors. Overall, the work connects AG-code flag geometry, divisor theory, and explicit constructions to yield self-orthogonal flags with potential quantum coding applications and insights into the structure of isometry vectors across function fields.

Abstract

A flag of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy , and it is said to satisfy the isometry-dual property with respect to an isometry vector if for . We characterize complete (i.e. ) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of . 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.
Paper Structure (6 sections, 9 theorems, 78 equations, 1 figure)

This paper contains 6 sections, 9 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The isometry-dual property in terms of building bases
  3. Translation of the isometry-dual property
  4. Construction of self-orthogonal flags of codes
  5. Characterizing the divisors giving the isometry-dual property
  6. Acknowledgments

Key Result

Lemma 2.1

For $0\leq i\leq m$ let $C_i$ be a linear code with length $n$, dimension $k_i$, and linear basis $u^i_1,u^i_2,\dots u^i_{k_i}$ in $\mathbb{F}_{q}^n$. A vector $x\in{\mathbb F}_q^n$ (not necessarily with nonzero coordinates) satisfies $C_{m-i}^\perp=x* C_i$ for all $i$ with $0\leq i\leq m$ if and on

Figures (1)

  • Figure 1: Klein function field over ${\mathbb F}_{ 8 }$ defined by the equation $X^3Y + Y^3Z + XZ^3 =0$. Analysis of flags $S_\beta$ satisfying the isometry-dual property. In this case $n= 21$, $g= 3$. We take $P=P_\infty=( 1 : 0 : 0 )$, $Q_1=( 0 : 1 : 0 )$, $Q_2=( 0 : 0 : 1 )$, and analyze the codes $C_{{\mathcal{L}}}(D,aP_\infty+\beta_1Q_1+\beta_2Q_2)$ with $\beta_2= 3$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 14 more