MinT - Best Known (219−103, 219, s)-Nets in Base 3

Best Known (219−103, 219, s)-Nets in Base 3

(219−103, 219, 78)-Net over F₃ — Constructive and digital

Digital (116, 219, 78)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 77, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (39, 142, 42)-net over F₃, using
  - net from sequence [i] based on digital (39, 41)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(219−103, 219, 129)-Net over F₃ — Digital

Digital (116, 219, 129)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−103, 219, 1037)-Net in Base 3 — Upper bound on s

There is no (116, 219, 1038)-net in base 3, because

1 times m-reduction [i] would yield (116, 218, 1038)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104 886993 061210 484985 253963 945581 162484 180715 542428 045886 665733 276845 948589 507587 857906 046968 744717 943777 > 3²¹⁸ [i]