MinT - Best Known (99−55, 99, s)-Nets in Base 27

Best Known (99−55, 99, s)-Nets in Base 27

(99−55, 99, 176)-Net over F₂₇ — Constructive and digital

Digital (44, 99, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 34, 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 (10, 65, 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]

(99−55, 99, 330)-Net over F₂₇ — Digital

Digital (44, 99, 330)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−55, 99, 370)-Net in Base 27 — Constructive

(44, 99, 370)-net in base 27, using

t-expansion [i] based on (43, 99, 370)-net in base 27, using
- 9 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]

(99−55, 99, 65865)-Net in Base 27 — Upper bound on s

There is no (44, 99, 65866)-net in base 27, because

1 times m-reduction [i] would yield (44, 98, 65866)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 780838 337369 910047 596804 798458 672994 997237 688898 879220 392054 133918 628314 845544 454202 586988 942411 550204 045984 107033 929372 420914 194978 653489 > 27⁹⁸ [i]