Covering Radius of Generalized Zetterberg Type Codes Over Finite Fields of Odd Characteristic

Abstract

Let Fq0 be a finite field of odd characteristic. For an integer s≥1 , let Cs(q0) be the generalized Zetterberg code of length qs0+1 over Fq0 . If s is even, then we prove that the covering radius of Cs(q0) is 3. Put q=qs0 . If s is odd and q≢7mod8 , then we present an explicit lower bound N1(q0) so that if s≥N1(q0) , then the covering radius of Cs(q0) is 3. We also show that the covering radius of C1(q0) is 2. Moreover we study some cases when s is an odd integer with 3≤s≤N1(q0) and, rather unexpectedly, we present concrete examples with covering radius 2 in that range. We introduce half generalized Zetterberg codes of length (qs0+1)/2 if q≡1mod4 . Similarly we introduce twisted half generalized Zetterberg codes of length (qs0+1)/2 if q≡3mod4 . We show that the same results hold for the half and twisted half generalized Zetterberg codes.