MinT - Best Known (94−51, 94, s)-Nets in Base 27

Best Known (94−51, 94, s)-Nets in Base 27

(94−51, 94, 178)-Net over F₂₇ — Constructive and digital

Digital (43, 94, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 33, 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]
2. digital (10, 61, 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]

(94−51, 94, 363)-Net over F₂₇ — Digital

Digital (43, 94, 363)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(94−51, 94, 370)-Net in Base 27 — Constructive

(43, 94, 370)-net in base 27, using

14 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]

(94−51, 94, 82652)-Net in Base 27 — Upper bound on s

There is no (43, 94, 82653)-net in base 27, because

1 times m-reduction [i] would yield (43, 93, 82653)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 087479 110355 637036 651556 106850 186490 951167 843089 879503 754153 700426 548390 710514 810946 729312 987676 483680 369950 574704 880539 252148 372563 > 27⁹³ [i]