MinT - Best Known (41, 41+51, s)-Nets in Base 27

Best Known (41, 41+51, s)-Nets in Base 27

(41, 41+51, 170)-Net over F₂₇ — Constructive and digital

Digital (41, 92, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 32, 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]
2. digital (9, 60, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(41, 41+51, 315)-Net over F₂₇ — Digital

Digital (41, 92, 315)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+51, 370)-Net in Base 27 — Constructive

(41, 92, 370)-net in base 27, using

8 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+51, 63493)-Net in Base 27 — Upper bound on s

There is no (41, 92, 63494)-net in base 27, because

1 times m-reduction [i] would yield (41, 91, 63494)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17956 483201 465776 631311 927002 059074 706909 464679 882362 395182 147059 719378 602438 895189 458189 272952 515168 066314 394214 271469 352487 783661 > 27⁹¹ [i]