MinT - Best Known (238−111, 238, s)-Nets in Base 3

Best Known (238−111, 238, s)-Nets in Base 3

(238−111, 238, 85)-Net over F₃ — Constructive and digital

Digital (127, 238, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 82, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (45, 156, 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]

(238−111, 238, 142)-Net over F₃ — Digital

Digital (127, 238, 142)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−111, 238, 1160)-Net in Base 3 — Upper bound on s

There is no (127, 238, 1161)-net in base 3, because

1 times m-reduction [i] would yield (127, 237, 1161)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 124430 311768 904422 235260 071627 954886 685312 449883 990572 840914 983389 360256 047965 108496 815019 098065 036527 300742 600427 > 3²³⁷ [i]