# From ds-bounds for cyclic codes to true distance for abelian codes

## TL;DR

This paper introduces a method to extend bounds on the minimum distance of cyclic codes to abelian codes using the concept of $ ext{B}$-apparent distance, improving bounds and identifying conditions for exact distances.

## Contribution

It develops a novel technique to generalize ds-bounds from cyclic to abelian codes and provides criteria for when the apparent distance equals the true minimum distance.

## Key findings

- Enhanced bounds for abelian code distances
- Conditions for $ ext{B}$-apparent distance to match true minimum distance
- Constructed tables of codes with verified minimum distances

## Abstract

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of $\mathbb{B}$-apparent distance. We use this technique to improve the searching for new bounds for the minimum distance of abelian codes. We also study conditions for an abelian code to verify that its $\mathbb{B}$-apparent distance reaches its (true) minimum distance. Then we construct some tables of such codes as an application