MinT - Best Known (37, 37+57, s)-Nets in Base 27

Best Known (37, 37+57, s)-Nets in Base 27

(37, 37+57, 132)-Net over F₂₇ — Constructive and digital

Digital (37, 94, 132)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 32, 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 (5, 62, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(37, 37+57, 224)-Net in Base 27 — Constructive

(37, 94, 224)-net in base 27, using

2 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]

(37, 37+57, 244)-Net over F₂₇ — Digital

Digital (37, 94, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 94, 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]

(37, 37+57, 298)-Net in Base 27

(37, 94, 298)-net in base 27, using

6 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]

(37, 37+57, 24657)-Net in Base 27 — Upper bound on s

There is no (37, 94, 24658)-net in base 27, because

1 times m-reduction [i] would yield (37, 93, 24658)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 095385 705970 443613 284059 894686 096793 377279 454652 138425 175965 228153 640911 777851 483924 086813 133789 856553 571885 622354 685519 472379 574809 > 27⁹³ [i]