We construct quantum MDS codes with parameters \( [\![ q^2+1,q^2+3-2d,d ]\!] _q\) for all \(d \leqslant q+1\), \(d \ne q\). These codes are shown to exist by proving that there are classical generalised Reed–Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if \(d\geqslant q+2\) then there is no generalised Reed–Solomon \([n,n-d+1,d]_{q^2}\) code which contains its Hermitian dual. We also construct an \( [\![ 18,0,10 ]\!] _5\) quantum MDS code, an \( [\![ 18,0,10 ]\!] _7\) quantum MDS code and a \( [\![ 14,0,8 ]\!] _5\) quantum MDS code, which are the first quantum MDS codes discovered for which \(d \geqslant q+3\), apart from the \( [\![ 10,0,6 ]\!] _3\) quantum MDS code derived from Glynn’s code.

我们为所有\ {d \ leqslant q + 1 \），\ {构造带有参数\（[\！[q ^ 2 + 1，q ^ 2 + 3-2d，d] \！] _q \）的量子MDS代码d \ ne q \）。通过证明存在包含其厄密对偶的经典广义Reed-Solomon码，可以证明这些代码存在。这些构造包括许多先前已知的构造，但是在某些情况下，这些代码是新的。我们继续证明，如果\（d \ geqslant q + 2 \）那么没有广义的Reed–Solomon \（[n，n-d + 1，d] _ {q ^ 2} \）代码包含厄米双重。我们还构造了一个\（[\！[18,0,10] \！] _5 \）量子MDS代码，一个\（[\！[18,0,10] \！] _7 \）量子MDS代码和一个\（[\！[14,0,8] \！] _5 \）量子MDS代码，这是发现的第一个量子MDS代码，其\（d \ geqslant q + 3 \）除\（[ ！[10,0,6] \！] _3 \）量子MDS代码是从格林的代码派生而来的。