Covering Radius of Generalized Zetterberg type Codes over finite fields of odd Characteristic

• Published in: IEEE transactions on information theory 2023-11, Vol.69 (11), p.1-1
• Main Authors: Shi, Minjia, Helleseth, Tor, OOzbudak, Ferruh
• Format: Article
• Language: English
• Subjects: algebraic curve; Codes; covering radius; Decoding; Fields (mathematics); finite field; Indexes; Informatics; Integers; Linear codes; Lower bounds; Parity check codes; Testing; Zetterberg codes
• ISSN: 0018-9448; 1557-9654
• DOI: 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.