MinT - Best Known (33, 33+45, s)-Nets in Base 27

Best Known (33, 33+45, s)-Nets in Base 27

(33, 33+45, 146)-Net over F₂₇ — Constructive and digital

Digital (33, 78, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 26, 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 (7, 52, 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]

(33, 33+45, 222)-Net over F₂₇ — Digital

Digital (33, 78, 222)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(33, 33+45, 224)-Net in Base 27 — Constructive

(33, 78, 224)-net in base 27, using

2 times m-reduction [i] based on (33, 80, 224)-net in base 27, using
- base change [i] based on digital (13, 60, 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]

(33, 33+45, 298)-Net in Base 27

(33, 78, 298)-net in base 27, using

6 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]

(33, 33+45, 35605)-Net in Base 27 — Upper bound on s

There is no (33, 78, 35606)-net in base 27, because

1 times m-reduction [i] would yield (33, 77, 35606)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 164 134961 496681 782251 000104 575274 502780 321762 775941 797120 737540 186472 569957 637147 458608 756416 315322 201359 171477 > 27⁷⁷ [i]