MinT - Best Known (59, 59+41, s)-Nets in Base 27

Best Known (59, 59+41, s)-Nets in Base 27

(59, 59+41, 264)-Net over F₂₇ — Constructive and digital

Digital (59, 100, 264)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (9, 29, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
3. digital (10, 51, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(59, 59+41, 434)-Net in Base 27 — Constructive

(59, 100, 434)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(59, 59+41, 2317)-Net over F₂₇ — Digital

Digital (59, 100, 2317)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+41, 3886712)-Net in Base 27 — Upper bound on s

There is no (59, 100, 3886713)-net in base 27, because

1 times m-reduction [i] would yield (59, 99, 3886713)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5070 079282 584414 501573 673142 989510 525162 601786 660796 567703 950131 600564 914444 442074 876302 523446 884522 170884 157293 806021 537037 202677 724249 630449 > 27⁹⁹ [i]