MinT - Best Known (239−65, 239, s)-Nets in Base 3

Best Known (239−65, 239, s)-Nets in Base 3

(239−65, 239, 288)-Net over F₃ — Constructive and digital

Digital (174, 239, 288)-net over F₃, using

t-expansion [i] based on digital (173, 239, 288)-net over F₃, using
- 4 times m-reduction [i] based on digital (173, 243, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 81, 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]

(239−65, 239, 720)-Net over F₃ — Digital

Digital (174, 239, 720)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−65, 239, 22585)-Net in Base 3 — Upper bound on s

There is no (174, 239, 22586)-net in base 3, because

1 times m-reduction [i] would yield (174, 238, 22586)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 358925 262364 479588 546142 617831 532902 709726 772079 672083 573091 190055 555502 817059 012102 132885 457227 075527 731325 784897 > 3²³⁸ [i]