MinT - Best Known (227−63, 227, s)-Nets in Base 3

Best Known (227−63, 227, s)-Nets in Base 3

(227−63, 227, 288)-Net over F₃ — Constructive and digital

Digital (164, 227, 288)-net over F₃, using

t-expansion [i] based on digital (163, 227, 288)-net over F₃, using
- 1 times m-reduction [i] based on digital (163, 228, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 76, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]

(227−63, 227, 641)-Net over F₃ — Digital

Digital (164, 227, 641)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−63, 227, 18649)-Net in Base 3 — Upper bound on s

There is no (164, 227, 18650)-net in base 3, because

1 times m-reduction [i] would yield (164, 226, 18650)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 675984 127398 290053 696020 235450 354388 619764 594691 718729 143887 817539 560438 833721 441757 743386 453986 619098 052633 > 3²²⁶ [i]