MinT - Best Known (243−109, 243, s)-Nets in Base 3

Best Known (243−109, 243, s)-Nets in Base 3

(243−109, 243, 86)-Net over F₃ — Constructive and digital

Digital (134, 243, 86)-net over F₃, using

5 times m-reduction [i] based on digital (134, 248, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 89, 38)-net over F₃, using
    - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 159, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(243−109, 243, 161)-Net over F₃ — Digital

Digital (134, 243, 161)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−109, 243, 1388)-Net in Base 3 — Upper bound on s

There is no (134, 243, 1389)-net in base 3, because

1 times m-reduction [i] would yield (134, 242, 1389)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 604687 332288 184366 040821 895432 123848 406386 457157 229688 402065 394255 748827 509813 631680 281976 854814 781617 547635 232017 > 3²⁴² [i]