MinT - Best Known (41, 41+29, s)-Nets in Base 27

Best Known (41, 41+29, s)-Nets in Base 27

(41, 41+29, 228)-Net over F₂₇ — Constructive and digital

Digital (41, 70, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 15, 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]
2. digital (6, 20, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 35, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(41, 41+29, 370)-Net in Base 27 — Constructive

(41, 70, 370)-net in base 27, using

30 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 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]

(41, 41+29, 1660)-Net over F₂₇ — Digital

Digital (41, 70, 1660)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+29, 2636706)-Net in Base 27 — Upper bound on s

There is no (41, 70, 2636707)-net in base 27, because

1 times m-reduction [i] would yield (41, 69, 2636707)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 580 899443 246146 025525 240311 166799 314651 550245 298909 175994 970277 879478 985946 912015 211573 520919 895229 > 27⁶⁹ [i]