MinT - Best Known (39, 39+48, s)-Nets in Base 27

Best Known (39, 39+48, s)-Nets in Base 27

(39, 39+48, 166)-Net over F₂₇ — Constructive and digital

Digital (39, 87, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 31, 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 (8, 56, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(39, 39+48, 310)-Net over F₂₇ — Digital

Digital (39, 87, 310)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+48, 370)-Net in Base 27 — Constructive

(39, 87, 370)-net in base 27, using

5 times m-reduction [i] based on (39, 92, 370)-net in base 27, using
- base change [i] based on digital (16, 69, 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]

(39, 39+48, 58209)-Net in Base 27 — Upper bound on s

There is no (39, 87, 58210)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33792 567660 459042 344805 136730 821425 874528 150988 037986 651440 652105 415979 951720 863375 447449 827655 842090 374042 586820 974329 879793 > 27⁸⁷ [i]