MinT - Best Known (33, 76, s)-Nets in Base 27

Best Known (33, 76, s)-Nets in Base 27

(33, 76, 152)-Net over F₂₇ — Constructive and digital

Digital (33, 76, 152)-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 (6, 49, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(33, 76, 224)-Net in Base 27 — Constructive

(33, 76, 224)-net in base 27, using

4 times m-reduction [i] based on (33, 80, 224)-net in base 27, using
- base change [i] based on digital (13, 60, 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]

(33, 76, 242)-Net over F₂₇ — Digital

Digital (33, 76, 242)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(33, 76, 298)-Net in Base 27

(33, 76, 298)-net in base 27, using

8 times m-reduction [i] based on (33, 84, 298)-net in base 27, using
- base change [i] based on digital (12, 63, 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]

(33, 76, 43194)-Net in Base 27 — Upper bound on s

There is no (33, 76, 43195)-net in base 27, because

1 times m-reduction [i] would yield (33, 75, 43195)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 225154 669918 782376 146126 947203 773553 208171 288574 582058 153419 920041 555363 150017 315012 769365 614635 805267 748407 > 27⁷⁵ [i]