MinT - Best Known (82−47, 82, s)-Nets in Base 27

Best Known (82−47, 82, s)-Nets in Base 27

(82−47, 82, 152)-Net over F₂₇ — Constructive and digital

Digital (35, 82, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 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, 53, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(82−47, 82, 224)-Net in Base 27 — Constructive

(35, 82, 224)-net in base 27, using

6 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]

(82−47, 82, 239)-Net over F₂₇ — Digital

Digital (35, 82, 239)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−47, 82, 298)-Net in Base 27

(35, 82, 298)-net in base 27, using

10 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]

(82−47, 82, 39833)-Net in Base 27 — Upper bound on s

There is no (35, 82, 39834)-net in base 27, because

1 times m-reduction [i] would yield (35, 81, 39834)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 239316 043479 724810 535271 136933 182246 090154 915210 245639 202839 903543 892977 542237 457267 370949 680691 225664 798271 673049 > 27⁸¹ [i]