Determining the Covering Radius of All Generalized Zetterberg Codes in Odd Characteristic

For an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s\ge 1$ </tex-math></inline-formula>, let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}_{s}(q_{0})$ </tex-math></inline-formula> be the generalized Zetterberg code of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{0}^{s}+1$ </tex-math></inline-formula> over the finite field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q_{0}}$ </tex-math></inline-formula> of odd characteristic. Recently, Shi et al. determined the covering radius of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {C}}_{s}(q_{0})$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{0}^{s} \cancel {\equiv }7 \pmod {8}$ </tex-math></inline-formula>, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{0}^{s} \equiv 7 \pmod {8}$ </tex-math></inline-formula>, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q_{0}^{s}+1}{2}$ </tex-math></inline-formula>, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.