MinT - Best Known (86−51, 86, s)-Nets in Base 27

Best Known (86−51, 86, s)-Nets in Base 27

(86−51, 86, 140)-Net over F₂₇ — Constructive and digital

Digital (35, 86, 140)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 29, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (6, 57, 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]

(86−51, 86, 220)-Net over F₂₇ — Digital

Digital (35, 86, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 86, 220)-net over F₂₇, using
- net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
    - Drinfeld modules of rank 1 [i]

(86−51, 86, 224)-Net in Base 27 — Constructive

(35, 86, 224)-net in base 27, using

2 times m-reduction [i] based on (35, 88, 224)-net in base 27, using
- base change [i] based on digital (13, 66, 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]

(86−51, 86, 298)-Net in Base 27

(35, 86, 298)-net in base 27, using

6 times m-reduction [i] based on (35, 92, 298)-net in base 27, using
- base change [i] based on digital (12, 69, 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]

(86−51, 86, 28780)-Net in Base 27 — Upper bound on s

There is no (35, 86, 28781)-net in base 27, because

1 times m-reduction [i] would yield (35, 85, 28781)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 46 368498 011967 706420 148216 070535 751207 471801 564369 509685 088788 612268 219611 384057 932638 910458 693046 882597 956149 597162 758643 > 27⁸⁵ [i]