MinT - Best Known (44−18, 44, s)-Nets in Base 27

Best Known (44−18, 44, s)-Nets in Base 27

(44−18, 44, 184)-Net over F₂₇ — Constructive and digital

Digital (26, 44, 184)-net over F₂₇, using

1 times m-reduction [i] based on digital (26, 45, 184)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 9, 56)-net over F₂₇, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 3, 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. digital (0, 6, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
  2. digital (4, 13, 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]
  3. digital (4, 23, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(44−18, 44, 266)-Net in Base 27 — Constructive

(26, 44, 266)-net in base 27, using

base change [i] based on digital (15, 33, 266)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (2, 11, 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]
  2. digital (4, 22, 150)-net over F₈₁, using
    - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(44−18, 44, 1407)-Net over F₂₇ — Digital

Digital (26, 44, 1407)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(44−18, 44, 1586921)-Net in Base 27 — Upper bound on s

There is no (26, 44, 1586922)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 955 010248 336754 816564 932327 535919 578286 236436 748964 747240 081653 > 27⁴⁴ [i]