Abstract
Let R = F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + vF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uvF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , with u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = u, v <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = v, uv = vu, where q = p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> for a positive integer m and an odd prime p. We study the algebraic structure of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes of block length (r, s). These codes can be viewed as R[x]-submodules of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> [x]/(x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> - 1) × R[x]/(x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - 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 F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes and construct some good QECCs.
Highlights
One of the important class of linear codes is that of cyclic codes
Motivated by the study of cyclic codes over mixed alphabets, we look at the structure of FqR-cyclic codes and their application in constructing quantum error-correcting codes (QECCs, in short)
5, we study the duality of FqR-cyclic codes and determine their generator polynomials
Summary
One of the important class of linear codes is that of cyclic codes. These codes have a significant role in the algebraic coding theory. Additive cyclic codes and described their generator polynomials and minimal generating sets They constructed several optimal and MDS codes from their study. Using the theory of mixed alphabets, Borges et al [13] introduced double cyclic codes over Z2 and determined their generator polynomials and spanning sets. The relationship between generator polynomials of double cyclic codes over Z2 and their duals was established They constructed many optimal binary codes from their study. Wu et al [45] studied triple cyclic codes over Z4 and obtained some new optimal linear codes over Z4 form their study In this line, recently, Dinh et al [23] discussed cyclic codes over mixed alphabets and studied their applications in constructing new quantum codes and LCD codes.
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