MinT - Best Known (91−38, 91, s)-Nets in Base 27

Best Known (91−38, 91, s)-Nets in Base 27

(91−38, 91, 248)-Net over F₂₇ — Constructive and digital

Digital (53, 91, 248)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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 (7, 26, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (8, 46, 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]

(91−38, 91, 398)-Net in Base 27 — Constructive

(53, 91, 398)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
2. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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]

(91−38, 91, 1886)-Net over F₂₇ — Digital

Digital (53, 91, 1886)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−38, 91, 2186348)-Net in Base 27 — Upper bound on s

There is no (53, 91, 2186349)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17951 716924 332675 957822 381906 222556 631422 796912 570863 793179 212523 938016 087476 922551 292482 723792 319729 978446 878174 963228 552048 377267 > 27⁹¹ [i]