A class of skew cyclic codes and application in quantum codes construction

Abstract

Let $p$ be an odd prime, and $k$ an integer such that $k\mid (p-1)$. This paper studies quantum codes from skew cyclic codes over the ring $R=\frac{{\mathbb{F}}_q\left[u\right]}{〈u^{k+1}-u〉}$, where $q$ is a power of the prime $p$. We construct a set of orthogonal idempotents of the ring $R$ and using the set, skew cyclic codes over the ring $R$ are decomposed as direct sum of skew cyclic codes over ${\mathbb{F}}_q$. We obtain a necessary and sufficient condition for a skew cyclic code to contain its dual. As an application, we construct quantum codes from skew cyclic codes over ${\mathbb{F}}_q$. It is observed that some quantum codes we constructed are MDS quantum codes.

Introduction

Classical computers work by manipulating bits that exist in one of two states, namely, 0 or 1. Quantum computers are not just limited to these two states, they are encoded with quantum data in the two conditions of 0 and 1 as quantum bits, or qubits, which can exist in superposition. That means, the qubits are both 0 and 1 and all points in between, which are processed at the same time, giving quantum computers the ability of performing many calculations at once. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Since quantum computers can contain these multiple states simultaneously, they have the potential to be millions of times more powerful than the most powerful classical supercomputers. Quantum computers will make use of qubits to encode quantum data and figure complex scientific issues utilizing the resources unique to quantum computers, such as direct access to superposition and entanglement. Using quantum computing, one can harness the magnificent powers superposition and entanglement to tackle complicated issues that classical computer systems cannot practically do. The two properties of superposition and entanglement together will empower qubits to process huge amounts of information at the same time and take care of complex issues. While traditional classical computer systems would need to arrange and figure out each conceivable arrangement, which may take a huge amount of time on a massive scale problem, quantum computers can locate every single imaginable variation simultaneously utilizing superposition and entanglement and move through a lot of information in an altogether limited quantity of time. As a quantum computer has the potential to simulate things that a classical computer could not, quantum computers outrun the classical computers in their ability to solve complex problems, and the application of error-correcting codes in quantum computers can be labeled as one of the pivotal reasons for this efficiency. Consequently, it has become evident that quantum error-correcting codes can protect quantum information investigations, and researches concerning quantum error-correcting codes have seen a tremendous headway. During the last few years, research on error-correcting codes has increased in both the public and private sectors. For instance, in October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), claimed to be able to perform a quantum computation that is infeasible on any classical computer.

Quantum error-correction is used in quantum computing to protect quantum information from errors which may occur due to decoherence and other quantum noise. Decoherence can revert a quantum system back to the classical system through interactions with the environment such as heat, light, sound, vibration, etc., which decay and eliminate the quantum behavior of particles. In 1995, Shor [66] introduced quantum error-correcting codes for the first time. In the next year, Steane [68] studied the properties of simple quantum error-correcting codes. However, it was after Calderbank et al. [19] that the construction of quantum error correcting codes from classical error-correcting codes started evolving rapidly. Several quantum error-correcting codes were constructed using cyclic codes over finite fields ${\mathbb{F}}_q$ (where $q$ is a power of prime number) with distinctive orthogonal properties, and as a result, numerous commendably good error correcting codes have been constructed [3], [42], [45], [46], [47], [49], [53], [56], [57], [58] over finite fields. Recently, the study of quantum error correcting codes over finite rings has got the attention of researchers, and many good quantum error correcting codes have been found [4], [5], [6], [7], [9], [10], [11], [12], [27], [28], [29], [44], [52], [64], [65]. From the last two decades, the study of quantum codes from algebraic curves [13], [26], [50], [54], [55] has also been developing rapidly.

The class of cyclic codes is a very important class of linear codes from both theoretical and practical point of view, which are easy to implement due to their rich algebraic structure. Cyclic codes have been extensively studied for the past six decades. Since the publication of a landmark paper by Hammons et al. in 1994 [48], which discovered that many well known good non-linear codes over ${\mathbb{F}}_2$ can be viewed as binary images under a Gray map of linear cyclic codes over ${\mathbb{Z}}_4$, there have been a lot of research on cyclic codes and their generalizations over finite rings (see, for example, [20], [21], [22], [23], [24], [25], [30], [32], [33], [34], [35], [36], [37], [38], [39], [40], [59]). However, all these works are restricted to codes whose alphabets are commutative rings.

In 1933, Ore [62] introduced a polynomial ring with usual polynomial addition and a specific multiplication to obtain a non-commutative structure over the polynomial ring, called skew polynomial ring. The non-commutative nature and rich algebraic structure of this class of rings allowed to construct many codes than the commutative set up, and so the study of codes over non-commutative polynomial rings has seen a rapid growth in recent time. In 2007, Boucher et al. [17], [18] studied skew cyclic codes over a skew polynomial ring. Later, skew quasi cyclic codes were introduced by Abualrab [2] and Bhaintwal [14]. In a few years after that, many authors studied some important structural properties of skew cyclic codes; see [1], [8], [43], [51], [67].

Motivated by that, in this paper, we study skew cyclic codes over the ring $R={\mathbb{F}}_q+u{\mathbb{F}}_q+\cdots +u^k{\mathbb{F}}_q$, where $u^{k+1}=u$ and $q=p^m$ for odd prime $p$ such that $k\mid (p-1)$, and then we obtain new quantum codes from the skew cyclic codes we constructed. The paper is organized as follows. Section 3 consists of the construction of linear codes and basic properties of Gray map over the ring $R$. In Section 4, we study the properties of skew cyclic codes and exhibit the generators of skew cyclic and dual skew cyclic codes over $R$. In Section 5, we give necessary and sufficient conditions for a skew cyclic code over $R$ to contain its dual code. Finally, as an application, we construct quantum codes from skew cyclic codes over $R$ via the Gray map defined over $R$. Section 6 concludes the paper.