MinT - Best Known (236−77, 236, s)-Nets in Base 3

Best Known (236−77, 236, s)-Nets in Base 3

(236−77, 236, 167)-Net over F₃ — Constructive and digital

Digital (159, 236, 167)-net over F₃, using

3¹ times duplication [i] based on digital (158, 235, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 47, 19)-net over F₃, using
    - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
        an explicitly constructive algebraic function field [i]
  2. digital (111, 188, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 94, 74)-net over F₉, using
      - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(236−77, 236, 393)-Net over F₃ — Digital

Digital (159, 236, 393)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−77, 236, 6667)-Net in Base 3 — Upper bound on s

There is no (159, 236, 6668)-net in base 3, because

1 times m-reduction [i] would yield (159, 235, 6668)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13328 731028 347111 129977 286134 728062 743868 304996 716395 624425 876699 781045 430460 625554 575465 471735 079692 036874 090505 > 3²³⁵ [i]