# Generalized Hadamard full propelinear codes

## TL;DR

This paper introduces a new class of codes called generalized Hadamard full propelinear codes, linking their existence to certain difference sets and exploring their structural properties with examples.

## Contribution

It establishes the equivalence between the existence of these codes and central relative difference sets, and studies their structural properties.

## Key findings

- Existence of generalized Hadamard full propelinear codes is equivalent to certain difference sets.
- Structural properties of these codes are characterized.
- Examples of these codes are provided.

## Abstract

Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative $(v,w,v,v/w)$-difference sets. Moreover, some structural properties of these codes are studied and examples are provided.