MinT - Best Known (77−43, 77, s)-Nets in Base 27

Best Known (77−43, 77, s)-Nets in Base 27

(77−43, 77, 158)-Net over F₂₇ — Constructive and digital

Digital (34, 77, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 27, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 50, 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]

(77−43, 77, 224)-Net in Base 27 — Constructive

(34, 77, 224)-net in base 27, using

7 times m-reduction [i] based on (34, 84, 224)-net in base 27, using
- base change [i] based on digital (13, 63, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(77−43, 77, 263)-Net over F₂₇ — Digital

Digital (34, 77, 263)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−43, 77, 298)-Net in Base 27

(34, 77, 298)-net in base 27, using

11 times m-reduction [i] based on (34, 88, 298)-net in base 27, using
- base change [i] based on digital (12, 66, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(77−43, 77, 50535)-Net in Base 27 — Upper bound on s

There is no (34, 77, 50536)-net in base 27, because

1 times m-reduction [i] would yield (34, 76, 50536)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 076656 502145 908161 631655 990786 181046 571917 613537 546263 474049 552505 387850 978925 079984 621531 553499 869548 580113 > 27⁷⁶ [i]