MinT - Best Known (90−55, 90, s)-Nets in Base 27

Best Known (90−55, 90, s)-Nets in Base 27

(90−55, 90, 128)-Net over F₂₇ — Constructive and digital

Digital (35, 90, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 31, 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, 59, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(90−55, 90, 172)-Net in Base 27 — Constructive

(35, 90, 172)-net in base 27, using

t-expansion [i] based on (34, 90, 172)-net in base 27, using
- 18 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
  - base change [i] based on digital (7, 81, 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]

(90−55, 90, 220)-Net over F₂₇ — Digital

Digital (35, 90, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 90, 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]

(90−55, 90, 298)-Net in Base 27

(35, 90, 298)-net in base 27, using

2 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]

(90−55, 90, 21945)-Net in Base 27 — Upper bound on s

There is no (35, 90, 21946)-net in base 27, because

1 times m-reduction [i] would yield (35, 89, 21946)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 625787 592213 454212 351983 075168 390312 613716 957068 290018 021486 173651 468260 788520 272206 361573 405999 501690 567447 575205 752742 151921 > 27⁸⁹ [i]