MinT - Best Known (247−117, 247, s)-Nets in Base 3

Best Known (247−117, 247, s)-Nets in Base 3

(247−117, 247, 85)-Net over F₃ — Constructive and digital

Digital (130, 247, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 85, 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, 162, 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]

(247−117, 247, 140)-Net over F₃ — Digital

Digital (130, 247, 140)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−117, 247, 1128)-Net in Base 3 — Upper bound on s

There is no (130, 247, 1129)-net in base 3, because

1 times m-reduction [i] would yield (130, 246, 1129)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2355 264350 345691 821412 501639 462030 203714 671640 343654 049954 887455 702235 683037 038700 744359 869723 505845 171214 762922 858105 > 3²⁴⁶ [i]