MinT - Best Known (161, 161+39, s)-Nets in Base 4

Best Known (161, 161+39, s)-Nets in Base 4

(161, 161+39, 1104)-Net over F₄ — Constructive and digital

Digital (161, 200, 1104)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (25, 44, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 22, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (117, 156, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 39, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(161, 161+39, 7405)-Net over F₄ — Digital

Digital (161, 200, 7405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(161, 161+39, 5344157)-Net in Base 4 — Upper bound on s

There is no (161, 200, 5344158)-net in base 4, because

1 times m-reduction [i] would yield (161, 199, 5344158)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645564 136250 297890 835415 238334 013008 441476 080866 801617 277875 820285 894987 932757 829035 659457 780503 814977 067648 241423 079798 > 4¹⁹⁹ [i]