MinT - Best Known (237−77, 237, s)-Nets in Base 3

Best Known (237−77, 237, s)-Nets in Base 3

(237−77, 237, 168)-Net over F₃ — Constructive and digital

Digital (160, 237, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 49, 20)-net over F₃, using
  - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [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]

(237−77, 237, 399)-Net over F₃ — Digital

Digital (160, 237, 399)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−77, 237, 6864)-Net in Base 3 — Upper bound on s

There is no (160, 237, 6865)-net in base 3, because

1 times m-reduction [i] would yield (160, 236, 6865)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40055 330468 116367 979092 222836 026763 346025 095633 752085 092034 095617 011380 943521 063292 608645 568916 361250 255989 919097 > 3²³⁶ [i]