MinT - Best Known (30, 65, s)-Nets in Base 27

Best Known (30, 65, s)-Nets in Base 27

(30, 65, 158)-Net over F₂₇ — Constructive and digital

Digital (30, 65, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 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, 42, 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]

(30, 65, 224)-Net in Base 27 — Constructive

(30, 65, 224)-net in base 27, using

3 times m-reduction [i] based on (30, 68, 224)-net in base 27, using
- base change [i] based on digital (13, 51, 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]

(30, 65, 289)-Net over F₂₇ — Digital

Digital (30, 65, 289)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 65, 298)-Net in Base 27

(30, 65, 298)-net in base 27, using

7 times m-reduction [i] based on (30, 72, 298)-net in base 27, using
- base change [i] based on digital (12, 54, 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]

(30, 65, 67542)-Net in Base 27 — Upper bound on s

There is no (30, 65, 67543)-net in base 27, because

1 times m-reduction [i] would yield (30, 64, 67543)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 40 484626 531700 852838 956818 962282 460285 066430 317275 689519 435412 088805 861864 222375 559021 584439 > 27⁶⁴ [i]