MinT - Best Known (83−50, 83, s)-Nets in Base 27

Best Known (83−50, 83, s)-Nets in Base 27

(83−50, 83, 128)-Net over F₂₇ — Constructive and digital

Digital (33, 83, 128)-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 (4, 54, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(83−50, 83, 172)-Net in Base 27 — Constructive

(33, 83, 172)-net in base 27, using

21 times m-reduction [i] based on (33, 104, 172)-net in base 27, using
- base change [i] based on digital (7, 78, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(83−50, 83, 220)-Net over F₂₇ — Digital

Digital (33, 83, 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]

(83−50, 83, 298)-Net in Base 27

(33, 83, 298)-net in base 27, using

1 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]

(83−50, 83, 22106)-Net in Base 27 — Upper bound on s

There is no (33, 83, 22107)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63576 463334 185817 776531 423474 254367 002465 498713 108250 062215 822095 763704 715721 904777 083109 149065 282625 117331 056573 592591 > 27⁸³ [i]