In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order $p^2^ where p is prime. The results obtained are applied to the dual of Hermitian repeated root cyclic codes to generate new quantum error-correcting codes.
Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.
Quantum error-correcting codes have attracted much attention. Among many research articles, the most general and systematic construction is the so called stabilizer code construction [6] or additive code construction [2], which constructs a quantum error-correcting code as an eigenspace of an Abelian subgroup S of the error group. Thereafter Calderbank et al. [3] proposed a construction of S from an additive code over the finite field F4 with 4 elements. These constructions work for tensor products of 2-state quantum systems. However Knill [8], [9] and Rains [13] observed that the construction [2], [6] can be generalized to n-state systems by an appropriate choice of the error basis. Rains [13] also generalized the construction [3] using additive codes over F4 to p-state quantum systems, but his generalization does not relate the problem of quantum code construction to classical error-correcting codes. We propose a construction of quantum error-correcting codes for p-state systems from classical error-correcting codes which is a generalization of [3]. Throughout this paper, p denotes a prime number andm a positive integer. This paper is organized as follows. In Sect. 2, we review the construction of quantum codes for nonbinary systems. In Sect. 3, we propose a construction of quantum codes for p-state systems from classical codes over Fp2 . In Sect. 4, we propose a construction of quantum codes for p-state systems from classical linear codes over Fp2m . In Sect. 5, we discuss a
Quantum codes constructed from cyclic codes over the ring Fq+vFq+v2Fq+v3Fq+v4Fq
In this article, we investigate properties of cyclic codes over a finite non-chain ring Fq+vFq+v2Fq+v3Fq+v4Fq, where q=pr,r is a positive integer, p is an odd prime, 4∣(p−1), and v5=v. As an application, we construct several quantum error correcting codes over the finite field Fq.
This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S, which is defined as S=Iq[u,v]/⟨u2−α2,v2−α2,uv−vu⟩, where Iq is a finite field of q elements, α be the nonzero elements of the field Iq and q is a power of an odd prime p such that q=pm,form≥1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S into a direct sum of constacyclic codes over Iq. We construct new and better quantum error-correcting codes over the ring S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).