MinT - Best Known (57, 57+49, s)-Nets in Base 27

Best Known (57, 57+49, s)-Nets in Base 27

(57, 57+49, 222)-Net over F₂₇ — Constructive and digital

Digital (57, 106, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 20, 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. digital (6, 30, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (7, 56, 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]

(57, 57+49, 370)-Net in Base 27 — Constructive

(57, 106, 370)-net in base 27, using

t-expansion [i] based on (43, 106, 370)-net in base 27, using
- 2 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(57, 57+49, 1068)-Net over F₂₇ — Digital

Digital (57, 106, 1068)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+49, 689601)-Net in Base 27 — Upper bound on s

There is no (57, 106, 689602)-net in base 27, because

1 times m-reduction [i] would yield (57, 105, 689602)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964295 155485 373017 159304 904808 173478 192526 809250 527728 980990 049103 695487 983469 263258 616514 885353 429115 312817 865565 961550 313549 358610 033370 983424 717553 > 27¹⁰⁵ [i]