MinT - Best Known (57−40, 57, s)-Nets in Base 27

Best Known (57−40, 57, s)-Nets in Base 27

(57−40, 57, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 57, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 57, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(57−40, 57, 116)-Net in Base 27 — Constructive

(17, 57, 116)-net in base 27, using

3 times m-reduction [i] based on (17, 60, 116)-net in base 27, using
- base change [i] based on digital (2, 45, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(57−40, 57, 144)-Net over F₂₇ — Digital

Digital (17, 57, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 57, 144)-net over F₂₇, using
- net from sequence [i] based on digital (16, 143)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 16 and N(F) ≥ 144, using
    - a curve from the manYPoints database [i]

(57−40, 57, 3824)-Net in Base 27 — Upper bound on s

There is no (17, 57, 3825)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3883 695543 080024 728546 180058 502821 920989 801815 041248 289371 666119 872979 101201 824561 > 27⁵⁷ [i]