Dual and Covering Radii of Extended Algebraic Geometry Codes

Many literatures consider the extended Reed-Solomon (RS) codes, including their dual codes and covering radii, but few focus on extended algebraic geometry (AG) codes of genus $g\ge1$. In this paper, we investigate extended AG codes and Roth-Lempel type AG codes, including their dual codes and minimum distances. Moreover, we show that for certain $g$, the length of a $g$-MDS code over a finite field $\mathbb{F}_q$ can attain $q+1+2g\sqrt{q}$, which is achieved by an extended AG code from the maximal curves of genus $g$. Notably, for some small finite fields, this length $q+1+2g\sqrt{q}$ is the largest among all known $g$-MDS codes. Subsequently, we establish that the covering radius of an $[n,k]$ extended AG code has $g+2$ possible values. For the case of $g=1$, we prove that this range reduces to two possible values when the length $n$ is sufficiently large, or when there exists an $[n,k+1]$ MDS elliptic code.