Quantum codes from trace codes

Boran Kim

Hao Chen

Md Ajaharul Hossain, Ramakrishna Bandi

Conjucyclic codes are part of a family of codes that includes cyclic, constacyclic, and quasi-cyclic codes, among others. Despite their importance in quantum error correction, they have not received much attention in the literature. This paper focuses on additive conjucyclic (ACC) codes over $ \mathbb{F}_4 $ and investigates their properties. Specifically, we derive the duals of ACC codes by using a trace inner product and obtain the trace hull and its dimension. Also, we establish a necessary and sufficient condition for an additive code to have a complementary dual (ACD). Additionally, we identify a necessary condition for an additive conjucyclic complementary pair of codes over $ \mathbb{F}_4 $. Furthermore, we show that the trace code of an ACC code is cyclic and provide a condition for the trace code of an ACC code to be LCD. To demonstrate the practical application of our findings, we construct some good entanglement-assisted quantum error-correcting (EAQEC) codes using the trace code of ACC codes.

Alexei Kitaev, Alexander Khanievich Shen', M. Vyalyi

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz +1 more

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature.

Sankalpa Khadka