MinT - Best Known (99−57, 99, s)-Nets in Base 27

Best Known (99−57, 99, s)-Nets in Base 27

(99−57, 99, 164)-Net over F₂₇ — Constructive and digital

Digital (42, 99, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 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 (7, 64, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(99−57, 99, 280)-Net over F₂₇ — Digital

Digital (42, 99, 280)-net over F₂₇, using

net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
  - Drinfeld modules of rank 1 [i]

(99−57, 99, 370)-Net in Base 27 — Constructive

(42, 99, 370)-net in base 27, using

5 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]

(99−57, 99, 44428)-Net in Base 27 — Upper bound on s

There is no (42, 99, 44429)-net in base 27, because

1 times m-reduction [i] would yield (42, 98, 44429)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 829913 725959 075022 907084 554036 504814 751323 660330 107863 716023 358828 977613 738342 919247 489243 425592 647209 453662 838431 993624 627378 298067 376017 > 27⁹⁸ [i]