MinT - Best Known (91−47, 91, s)-Nets in Base 27

Best Known (91−47, 91, s)-Nets in Base 27

(91−47, 91, 190)-Net over F₂₇ — Constructive and digital

Digital (44, 91, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 33, 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]
2. digital (11, 58, 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]

(91−47, 91, 370)-Net in Base 27 — Constructive

(44, 91, 370)-net in base 27, using

t-expansion [i] based on (43, 91, 370)-net in base 27, using
- 17 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]

(91−47, 91, 475)-Net over F₂₇ — Digital

Digital (44, 91, 475)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−47, 91, 144689)-Net in Base 27 — Upper bound on s

There is no (44, 91, 144690)-net in base 27, because

1 times m-reduction [i] would yield (44, 90, 144690)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 664 965128 731450 296787 639885 666542 038496 120078 792821 395398 390156 363862 410414 511400 157085 830742 209217 198137 868873 780188 622144 748729 > 27⁹⁰ [i]