Some constructions of quantum MDS codes

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