MinT - Best Known (40, 40+44, s)-Nets in Base 27

Best Known (40, 40+44, s)-Nets in Base 27

(40, 40+44, 178)-Net over F₂₇ — Constructive and digital

Digital (40, 84, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 30, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]
2. digital (10, 54, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(40, 40+44, 370)-Net in Base 27 — Constructive

(40, 84, 370)-net in base 27, using

12 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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]

(40, 40+44, 410)-Net over F₂₇ — Digital

Digital (40, 84, 410)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+44, 101633)-Net in Base 27 — Upper bound on s

There is no (40, 84, 101634)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716216 282933 887008 912538 258216 285023 376815 700865 668663 549050 756477 080778 047779 377887 961601 396185 607372 626992 608718 216477 > 27⁸⁴ [i]