MinT - Best Known (43, 43+33, s)-Nets in Base 27

Best Known (43, 43+33, s)-Nets in Base 27

(43, 43+33, 216)-Net over F₂₇ — Constructive and digital

Digital (43, 76, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 15, 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 (6, 22, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (6, 39, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(43, 43+33, 370)-Net in Base 27 — Constructive

(43, 76, 370)-net in base 27, using

32 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]

(43, 43+33, 1250)-Net over F₂₇ — Digital

Digital (43, 76, 1250)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(43, 43+33, 1339924)-Net in Base 27 — Upper bound on s

There is no (43, 76, 1339925)-net in base 27, because

1 times m-reduction [i] would yield (43, 75, 1339925)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 225052 531070 637985 914641 716493 266424 621484 816268 528097 824990 814987 301638 044777 929186 017075 809026 065125 989441 > 27⁷⁵ [i]