A Study of F_qR-Cyclic Codes and Their Applications in Constructing Quantum Codes

Abstract

Let R = Fq + uFq + vFq + uvFq, with u2 = u, v2 = v, uv = vu, where q = pm for a positive integer m and an odd prime p. We study the algebraic structure of FqR-cyclic codes of block length (r, s). These codes can be viewed as R[x]-submodules of Fq[x]/(xr - 1) × R[x]/(xs - 1). For this family of codes we discuss the generator polynomials and minimal generating sets. We study the algebraic structure of separable codes. Further, we discuss the duality of this family of codes and determine their generator polynomials. We obtain several optimal and near-optimal codes from this study. As applications, we discuss a construction of quantum error-correcting codes (QECCs) from FqR-cyclic codes and construct some good QECCs.

Keywords

• qR-cyclic codes%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> F<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">qR</italic>-cyclic codes
• generator polynomials
• minimal generating sets
• QECCs