MinT - Best Known (50, 50+43, s)-Nets in Base 27

Best Known (50, 50+43, s)-Nets in Base 27

(50, 50+43, 210)-Net over F₂₇ — Constructive and digital

Digital (50, 93, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 18, 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, 25, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 50, 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]

(50, 50+43, 370)-Net in Base 27 — Constructive

(50, 93, 370)-net in base 27, using

t-expansion [i] based on (43, 93, 370)-net in base 27, using
- 15 times m-reduction [i] based on (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]

(50, 50+43, 959)-Net over F₂₇ — Digital

Digital (50, 93, 959)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+43, 622655)-Net in Base 27 — Upper bound on s

There is no (50, 93, 622656)-net in base 27, because

1 times m-reduction [i] would yield (50, 92, 622656)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484692 635921 772053 624175 765174 526662 906733 152902 473319 658069 907622 455705 957467 342906 488614 848542 420858 523685 749893 323090 571432 812161 > 27⁹² [i]