MinT - Best Known (43, 43+65, s)-Nets in Base 27

Best Known (43, 43+65, s)-Nets in Base 27

(43, 43+65, 146)-Net over F₂₇ — Constructive and digital

Digital (43, 108, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 36, 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, 72, 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]

(43, 43+65, 280)-Net over F₂₇ — Digital

Digital (43, 108, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 108, 280)-net over F₂₇, using
- net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(43, 43+65, 370)-Net in Base 27 — Constructive

(43, 108, 370)-net in base 27, using

base change [i] based on digital (16, 81, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(43, 43+65, 30046)-Net in Base 27 — Upper bound on s

There is no (43, 108, 30047)-net in base 27, because

1 times m-reduction [i] would yield (43, 107, 30047)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1433 330626 700114 672236 943094 672792 635748 924692 921161 749894 719626 697395 515604 408133 651338 887809 029499 837322 401191 727110 412850 434829 982121 069433 942105 875649 > 27¹⁰⁷ [i]