MinT - Best Known (42, 42+53, s)-Nets in Base 27

Best Known (42, 42+53, s)-Nets in Base 27

(42, 42+53, 170)-Net over F₂₇ — Constructive and digital

Digital (42, 95, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 33, 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 (9, 62, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(42, 42+53, 312)-Net over F₂₇ — Digital

Digital (42, 95, 312)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+53, 370)-Net in Base 27 — Constructive

(42, 95, 370)-net in base 27, using

9 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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]

(42, 42+53, 60695)-Net in Base 27 — Upper bound on s

There is no (42, 95, 60696)-net in base 27, because

1 times m-reduction [i] would yield (42, 94, 60696)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 476830 472710 508588 342776 943528 203600 798755 129515 163939 118781 674169 861864 279593 434958 393076 460563 874639 666481 870421 670450 756950 445713 > 27⁹⁴ [i]