MinT - Best Known (90−53, 90, s)-Nets in Base 27

Best Known (90−53, 90, s)-Nets in Base 27

(90−53, 90, 146)-Net over F₂₇ — Constructive and digital

Digital (37, 90, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 30, 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, 60, 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]

(90−53, 90, 224)-Net in Base 27 — Constructive

(37, 90, 224)-net in base 27, using

6 times m-reduction [i] based on (37, 96, 224)-net in base 27, using
- base change [i] based on digital (13, 72, 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]

(90−53, 90, 244)-Net over F₂₇ — Digital

Digital (37, 90, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 90, 244)-net over F₂₇, using
- net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
    - Drinfeld modules of rank 1 [i]

(90−53, 90, 298)-Net in Base 27

(37, 90, 298)-net in base 27, using

10 times m-reduction [i] based on (37, 100, 298)-net in base 27, using
- base change [i] based on digital (12, 75, 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]

(90−53, 90, 32196)-Net in Base 27 — Upper bound on s

There is no (37, 90, 32197)-net in base 27, because

1 times m-reduction [i] would yield (37, 89, 32197)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 638921 784926 525367 974229 656762 632433 829672 716908 649488 319131 095570 238936 564575 180940 802459 373792 589568 680819 929425 122488 821697 > 27⁸⁹ [i]