MinT - Best Known (238−115, 238, s)-Nets in Base 3

Best Known (238−115, 238, s)-Nets in Base 3

(238−115, 238, 80)-Net over F₃ — Constructive and digital

Digital (123, 238, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 78, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 160, 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]

(238−115, 238, 128)-Net over F₃ — Digital

Digital (123, 238, 128)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−115, 238, 1008)-Net in Base 3 — Upper bound on s

There is no (123, 238, 1009)-net in base 3, because

1 times m-reduction [i] would yield (123, 237, 1009)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 123956 075894 516092 224015 490029 971161 484982 126027 372532 661812 819382 682927 010722 005115 715991 825997 401769 411423 827715 > 3²³⁷ [i]