MinT - Best Known (165−39, 165, s)-Nets in Base 3

Best Known (165−39, 165, s)-Nets in Base 3

(165−39, 165, 464)-Net over F₃ — Constructive and digital

Digital (126, 165, 464)-net over F₃, using

3¹ times duplication [i] based on digital (125, 164, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 41, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(165−39, 165, 905)-Net over F₃ — Digital

Digital (126, 165, 905)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−39, 165, 52040)-Net in Base 3 — Upper bound on s

There is no (126, 165, 52041)-net in base 3, because

1 times m-reduction [i] would yield (126, 164, 52041)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 770005 927816 707247 057117 020475 938606 650493 351786 706513 845123 265271 766027 964475 > 3¹⁶⁴ [i]