# Hybrid Codes

## TL;DR

Hybrid codes enable simultaneous encoding of classical and quantum information with enhanced error detection capabilities, introducing new weight enumerators that differ from traditional quantum codes.

## Contribution

The paper introduces the concept of hybrid codes, demonstrating their superior error detection and defining new weight enumerators that do not follow the MacWilliams identity.

## Key findings

- Hybrid codes can detect more errors than comparable quantum codes.
- Weight enumerators for hybrid codes are introduced and characterized.
- Hybrid codes' weight enumerators do not obey the MacWilliams identity.

## Abstract

A hybrid code can simultaneously encode classical and quantum information into quantum digits such that the information is protected against errors when transmitted through a quantum channel. It is shown that a hybrid code has the remarkable feature that it can detect more errors than a comparable quantum code that is able to encode the classical and quantum information. Weight enumerators are introduced for hybrid codes that allow to characterize the minimum distance of hybrid codes. Surprisingly, the weight enumerators for hybrid codes do not obey the usual MacWilliams identity.