Fq -linear codes over Fq -algebras

Tushar Bag, Daniel Panario

Xiusheng Liu, Hualu Liu

Hai Quang Dinh, Sachin Pathak, Tushar Bag +2 more

Let R = Fq+ uFq+ vFq+ uvFq, with u2= u, v2= v, uv = vu, where q = pmfor 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.

Kenny Barrese

Mohd Azmi, Mohammad Fareed Ahmad, Nadeem ur Rehman +1 more

Let q=pr, where p is an odd prime and r be a positive integer, 𝒮=𝔽q[u]〈u2−1〉 and 𝒯=𝔽q[u,v]〈u2−1,v2−1,uv=vu〉 and Ω=(ω1,ω2,ω3)∈𝔽q𝒮𝒯⁠. The algebraic structure of constacyclic codes over 𝔽q𝒮𝒯−Ω–constacyclic code of block length (l, m, n) is examined in this paper. The construction of quantum error-correcting codes (QECCs) from constacyclic codes over 𝔽q𝒮𝒯 is also covered in this paper and we derive some new QECCs.

Hai Quang Dinh, Tushar Bag, Ashish Kumar Upadhyay +2 more

Let p be an odd prime, q = pm, R = Fq+ uFqwith u2= 1, and S = Fq+ uFq+ vFq+ uvFqwith u2= 1, v2= 1, uv = vu. In this paper, FqRS-cyclic codes over FqRS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the FqRS-cyclic codes over FqRS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the FqRS-cyclic codes over FqRS. Among others, we construct a Gray map over FqRS and discuss the Gray images of FqRS-cyclic codes over Fq.