MinT - Best Known (43, 43+57, s)-Nets in Base 27

Best Known (43, 43+57, s)-Nets in Base 27

(43, 43+57, 166)-Net over F₂₇ — Constructive and digital

Digital (43, 100, 166)-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 (8, 65, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(43, 43+57, 289)-Net over F₂₇ — Digital

Digital (43, 100, 289)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(43, 43+57, 370)-Net in Base 27 — Constructive

(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]

(43, 43+57, 49980)-Net in Base 27 — Upper bound on s

There is no (43, 100, 49981)-net in base 27, because

1 times m-reduction [i] would yield (43, 99, 49981)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5072 207927 941926 556230 603886 768793 423395 397188 839600 249711 761271 527535 100070 453054 288315 914120 182736 491450 020921 840431 970879 429196 679231 975953 > 27⁹⁹ [i]