MDS Algebraic Geometry Codes of Square-Root Length on Hyperelliptic Curves

Abstract

For every fixed genus $g\ge 2$ and every $0<\rho <1$, we construct maximum distance separable (MDS) algebraic geometry codes over $\F_q$ for all sufficiently large odd squares $q$. Their parameters are $[n,\lfloor \rho n\rfloor ,n-\lfloor \rho n\rfloor +1{]}_q$ with $n=\sqrt{q}+1+O_g(q^{1/4})$. The codes use full Riemann--Roch spaces on genus-$g$ hyperelliptic curves. They are not monomially equivalent to generalized Reed--Solomon (GRS) codes when $q$ is sufficiently large. An infinite subfamily has exact length $n=\sqrt{q}+2g{q}^{1/4}+1$.

arXiv comment10 pages, 2 figures