Hulls of Reed-Solomon Codes via Algebraic Geometry Codes

Let ${\mathrm{ RS}}_{k}(\mathbf {a})$ be a $k$ -dimensional Reed-Solomon (RS) code over $\mathbb {F}_{q}$ associated with $\mathbf {a}=(\alpha _{1},\cdots,\alpha _{n})$ and let $h=\prod _{i=1}^{n}(z-\alpha _{i})$ be a polynomial in variable $z$ . In this paper, by expressing ${\mathrm{ RS}}_{k}(\mathbf {a})$ as an $\mathcal {L}$ -construction algebraic geometry code, we completely determine the dimension of the hull ${\mathrm{ RS}}_{k}(\mathbf {a})\bigcap {\mathrm{ RS}}_{k}(\mathbf {a})^{\perp} $ in terms of the degree of the derivative of $h$ and some relevant polynomials. As applications, we explicitly determine the parameters of MDS entanglement-assisted quantum error-correcting codes constructed from RS codes, and all linear complementary dual (resp. self-dual) RS codes are also fully described.