# Toric Codes, Multiplicative Structure and Decoding

## TL;DR

This paper explores the multiplicative structure of toric codes over finite fields, enabling decoding, secret sharing, and quantum error correction, thus extending classical coding theory into new algebraic and quantum domains.

## Contribution

It introduces the use of toric varieties' multiplicative structure for decoding, secret sharing, and quantum code construction, generalizing known methods from Reed-Solomon codes.

## Key findings

- Decoding of toric codes using their multiplicative structure.
- Construction of linear secret sharing schemes with strong multiplication.
- Development of quantum error correcting codes from toric surfaces.

## Abstract

Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for \emph{decoding}, resembling the decoding of Reed-Solomon codes and aligns with decoding by error correcting pairs. We have used the multiplicative structure on toric codes to construct linear secret sharing schemes with \emph{strong multiplication} via Massey's construction generalizing the Shamir Linear secret sharing shemes constructed from Reed-Solomon codes. We have constructed quantum error correcting codes from toric surfaces by the Calderbank-Shor-Steane method.