MinT - Best Known (59−37, 59, s)-Nets in Base 27

Best Known (59−37, 59, s)-Nets in Base 27

(59−37, 59, 112)-Net over F₂₇ — Constructive and digital

Digital (22, 59, 112)-net over F₂₇, using

net from sequence [i] based on digital (22, 111)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 22 and N(F) ≥ 112, using
  - an explicitly constructive algebraic function field [i]

(59−37, 59, 163)-Net over F₂₇ — Digital

Digital (22, 59, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 59, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(59−37, 59, 172)-Net in Base 27 — Constructive

(22, 59, 172)-net in base 27, using

1 times m-reduction [i] based on (22, 60, 172)-net in base 27, using
- base change [i] based on digital (7, 45, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(59−37, 59, 190)-Net in Base 27

(22, 59, 190)-net in base 27, using

5 times m-reduction [i] based on (22, 64, 190)-net in base 27, using
- base change [i] based on digital (6, 48, 190)-net over F₈₁, using
  - net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
      - a curve from the manYPoints database [i]

(59−37, 59, 11884)-Net in Base 27 — Upper bound on s

There is no (22, 59, 11885)-net in base 27, because

1 times m-reduction [i] would yield (22, 58, 11885)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 104521 917797 872496 179377 160338 559145 667766 100596 862504 355278 838960 586441 721010 053745 > 27⁵⁸ [i]