Covering Radius of Generalized Zetterberg Type Codes Over Finite Fields of Odd Characteristic
Minjia Shi, Tor Helleseth, Ferruh Özbudak · IEEE Transactions on Information Theory · 2023
Let$ {\mathbb F}_{q_{0}}$be a finite field of odd characteristic. For an integer$s\ge 1$, let$\mathcal {C}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over$ {\mathbb F}_{q_{0}}$. If$s$is even, then we prove that the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. Put$q=q_{0}^{s}$. If$s$is odd and$q ot \equiv 7 \mod 8$, then we present an explicit lower bound$N_{1}(q_{0})$so that if$s \ge N_{1}(q_{0})$, then the covering radius of$\mathcal {C}_{s}(q_{0})$is 3. We also show that the covering radius of$\mathcal {C}_{1}(q_{0})$is 2. Moreover we study some cases when$s$is an odd integer with$3 \le s \le N_{1}(q_{0})$and, rather unexpectedly, we present concrete examples with covering radius 2 in that range. We introduce half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 1 \mod 4$. Similarly we introduce twisted half generalized Zetterberg codes of length$(q_{0}^{s}+1)/2$if$q \equiv 3 \mod 4$. We show that the same results hold for the half and twisted half generalized Zetterberg codes.