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

TL;DR

This paper studies special structured sequences of linear codes called flags, characterizing their duality properties, especially for algebraic geometry codes, and provides constructions and classifications of such flags.

Contribution

It introduces a characterization of complete isometry-dual flags via linear subspaces and constructs self-orthogonal flags over maximal function fields.

Findings

Characterization of complete isometry-dual flags using linear subspaces.

A translation property of isometry-dual flags for algebraic geometry codes.

Finite classification of isometry vectors related to divisors in 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.