MinT - Best Known (89−51, 89, s)-Nets in Base 27

Best Known (89−51, 89, s)-Nets in Base 27

(89−51, 89, 158)-Net over F₂₇ — Constructive and digital

Digital (38, 89, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 31, 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, 58, 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]

(89−51, 89, 224)-Net in Base 27 — Constructive

(38, 89, 224)-net in base 27, using

11 times m-reduction [i] based on (38, 100, 224)-net in base 27, using
- base change [i] based on digital (13, 75, 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]

(89−51, 89, 255)-Net over F₂₇ — Digital

Digital (38, 89, 255)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−51, 89, 298)-Net in Base 27

(38, 89, 298)-net in base 27, using

15 times m-reduction [i] based on (38, 104, 298)-net in base 27, using
- base change [i] based on digital (12, 78, 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]

(89−51, 89, 42748)-Net in Base 27 — Upper bound on s

There is no (38, 89, 42749)-net in base 27, because

1 times m-reduction [i] would yield (38, 88, 42749)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912280 220735 342922 590185 848427 959157 459634 052370 848965 062948 150510 703480 990687 858957 843288 536887 106934 076950 507244 123889 553811 > 27⁸⁸ [i]