Covering radius of generalized Zetterberg type codes over finite fields of odd characteristic

Official URL: https://dx.doi.org/10.1109/TIT.2023.3296754

Let F q0 be a finite field of odd characteristic. For an integer s ≥ 1, let C s (q0) be the generalized Zetterberg code of length q s 0 + 1 over F q0 . If s is even, then we prove that the covering radius of C s (q0) is 3. Put q = q s 0 . If s is odd and q ≢ 7 mod 8, then we present an explicit lower bound N 1 (q0) so that if s ≥ N 1 (q0), then the covering radius of C s (q0) is 3. We also show that the covering radius of C 1 (q0) is 2. Moreover we study some cases when s is an odd integer with 3 ≤ s ≤ N 1 (q0) and, rather unexpectedly, we present concrete examples with covering radius 2 in that range. We introduce half generalized Zetterberg codes of length (q s 0 + 1)/2 if q ≡ 1 mod 4. Similarly we introduce twisted half generalized Zetterberg codes of length (q s 0 + 1)/2 if q ≡ 3 mod 4. We show that the same results hold for the half and twisted half generalized Zetterberg codes.