Hulls of Reed-Solomon Codes via Algebraic Geometry Codes
Bocong Chen, San Ling, Hongwei Liu · IEEE Transactions on Information Theory · 2022
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.