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We prove that any Hermitian self-orthogonal $[n,k,d]_{q^2}$ code gives riseto an $[n,k,d]_{q^2}$ code with $\ell$ dimensional Hermitian hull for $0\le\ell \le k$. We present a new method to construct Hermitian self-orthogonal$[n,k]_{q^2}$ codes with large dimensions $k>\frac{n+q-1}{q+1}$. New familiesof Hermitian self-orthogonal codes with good parameters are obtained; moreprecisely those containing almost MDS codes. By applying a puncturing techniqueto Hermitian self-orthogonal codes, MDS $[n,k]_{q^2}$ linear codes withHermitian hull having large dimensions $k>\frac{n+q-1}{q+1}$ are also derived.New families of MDS, almost MDS and optimal codes with arbitrary Hermitian hulldimensions are explicitly constructed from algebraic curves. As an application,we provide entanglement-assisted quantum error correcting codes with newparameters.

Explicit constructions of optimal linear codes with Hermitian hulls and their application to quantum codes