Quantum Cyclic Code of length dividing $p^{t}+1$

TL;DR

This paper introduces a new family of quantum cyclic stabiliser codes called Frobenius codes, characterizes their structure, and provides efficient decoding algorithms, including for non-linear codes, expanding the landscape of quantum error correction.

Contribution

It defines Frobenius codes, characterizes linear cases completely, and generalizes BCH decoding to a new class of non-linear quantum codes with efficient decoding.

Findings

Complete characterization of linear Frobenius codes.

Existence of non-linear Frobenius codes with efficient decoding.

Construction methods for all Frobenius codes when t is even.

Abstract

In this paper, we study cyclic stabiliser codes over $\mathbb{F}_p$ of length dividing $p^t+1$ for some positive integer $t$. We call these $t$-Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms. An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes. As a consequence, we show that for every integer $n$ that divides $p^t+1$ for an odd $t$, there are no linear cyclic codes of length $n$. On the other hand for even $t$, we give an explicit method to construct all of them. This gives us a many explicit example of Frobenius codes which include the well studied Laflamme code. We show that the classical notion of BCH distance can be generalised to all the Frobenius codes that we construct, including the non-linear ones, and show that the algorithm of Berlekamp can be generalised to correct quantum errors within the BCH limit. This gives, for the first time, a family of codes that are neither CSS nor linear for which efficient decoding algorithm exits. The explicit examples that we construct are summarised in Table \ref{tab:explicit-examples-short} and explained in detail in Tables \ref{tab:explicit-examples-2} (linear case) and \ref{tab:explicit-examples-3} (non-linear case).