MinT - Best Known (45−29, 45, s)-Nets in Base 27

Best Known (45−29, 45, s)-Nets in Base 27

(45−29, 45, 96)-Net over F₂₇ — Constructive and digital

Digital (16, 45, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 45, 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]

(45−29, 45, 144)-Net over F₂₇ — Digital

Digital (16, 45, 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]

(45−29, 45, 150)-Net in Base 27 — Constructive

(16, 45, 150)-net in base 27, using

3 times m-reduction [i] based on (16, 48, 150)-net in base 27, using
- base change [i] based on digital (4, 36, 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]

(45−29, 45, 154)-Net in Base 27

(16, 45, 154)-net in base 27, using

3 times m-reduction [i] based on (16, 48, 154)-net in base 27, using
- base change [i] based on digital (4, 36, 154)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 153)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 154, using
      - a curve from the manYPoints database [i]

(45−29, 45, 7322)-Net in Base 27 — Upper bound on s

There is no (16, 45, 7323)-net in base 27, because

1 times m-reduction [i] would yield (16, 44, 7323)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 956 543527 053852 910870 183256 131978 836219 584818 939171 161511 134189 > 27⁴⁴ [i]