Entanglement-assisted quantum error correction codes with length \(n=q^2+1\)

In this paper, by investigating \(q^2\)-cyclotomic coset modulo rn in detail, where q is a prime power, \(n=q^2+1\) and \(r\mid (q+1)\), series of entanglement-assisted quantum error correction (EAQEC) codes with flexible parameters of length n are constructed from constacyclic codes (including cyclic codes). Most of our EAQEC codes are new and have large minimum distance. As to EAQEC codes constructed from cyclic codes, their all possible parameters are determined completely. When minimum distance \(d\le \frac{n+2}{2}\), all of our constructed EAQEC codes are entanglement-assisted quantum MDS (EAQMDS) codes. Those previously known EAQMDS codes with the same length in Fan et al. (Quantum Inf Comput 16:423–434, 2016), Chen et al. (Quantum Inf Process 16(303):1–22, 2017), Lu et al. (Finite Fields Their Appl 53:309–325, 2018), Mustafa and Emre (Comput Appl Math 38(75):1–13, 2019) and Qian and Zhang (Quantum Inf Process 18(71):1–12, 2019) are special cases of ours. Besides, some maximum entanglement EAQEC codes and maximum entanglement EAQMDS codes are derived as well.

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