MinT - Best Known (100−53, 100, s)-Nets in Base 27

Best Known (100−53, 100, s)-Nets in Base 27

(100−53, 100, 190)-Net over F₂₇ — Constructive and digital

Digital (47, 100, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 36, 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, 64, 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]

(100−53, 100, 370)-Net in Base 27 — Constructive

(47, 100, 370)-net in base 27, using

t-expansion [i] based on (43, 100, 370)-net in base 27, using
- 8 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]

(100−53, 100, 437)-Net over F₂₇ — Digital

Digital (47, 100, 437)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−53, 100, 114409)-Net in Base 27 — Upper bound on s

There is no (47, 100, 114410)-net in base 27, because

1 times m-reduction [i] would yield (47, 99, 114410)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5070 886733 796357 799023 348623 357654 095482 179059 125771 965311 173205 861760 414204 594446 060557 597191 392458 764783 042092 829778 928967 898698 925966 464373 > 27⁹⁹ [i]