MinT - Best Known (82−48, 82, s)-Nets in Base 27

Best Known (82−48, 82, s)-Nets in Base 27

(82−48, 82, 140)-Net over F₂₇ — Constructive and digital

Digital (34, 82, 140)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 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, 54, 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]

(82−48, 82, 220)-Net over F₂₇ — Digital

Digital (34, 82, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 82, 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]

(82−48, 82, 224)-Net in Base 27 — Constructive

(34, 82, 224)-net in base 27, using

2 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]

(82−48, 82, 298)-Net in Base 27

(34, 82, 298)-net in base 27, using

6 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]

(82−48, 82, 29288)-Net in Base 27 — Upper bound on s

There is no (34, 82, 29289)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2354 959119 782364 655063 555537 084138 301521 691088 687696 566565 159536 932492 593883 941053 853013 254675 861476 687686 025345 020833 > 27⁸² [i]