# Hulls of Linear Codes Revisited with Applications

## TL;DR

This paper explores the algebraic properties of hulls of linear codes, providing new characterizations and applications, including constructions of quantum error-correcting codes, especially over fields of odd characteristic.

## Contribution

It introduces alternative characterizations of hulls of linear codes and links their properties to Gramian matrices, enabling new code constructions.

## Key findings

- Gramian of generator matrices over odd characteristic fields is diagonalizable.
- A linear code over odd characteristic fields is complementary dual iff it has an orthogonal basis.
- New constructions of entanglement-assisted quantum error-correcting codes are provided.

## Abstract

Hulls of linear codes have been of interest and extensively studied due to their rich algebraic structures and wide applications. In this paper, alternative characterizations of hulls of linear codes are given as well as their applications. Properties of hulls of linear codes are given in terms of their Gramians of their generator and parity-check matrices. Moreover, it is show that the Gramian of a generator matrix of every linear code over a finite field of odd characteristic is diagonalizable. Subsequently, it is shown that a linear code over a finite field of odd characteristic is complementary dual if and only if it has an orthogonal basis. Based on this characterization, constructions of good entanglement-assisted quantum error-correcting codes are provided.