MinT - Best Known (19, 42, s)-Nets in Base 27

Best Known (19, 42, s)-Nets in Base 27

(19, 42, 128)-Net over F₂₇ — Constructive and digital

Digital (19, 42, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 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]
2. digital (4, 27, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(19, 42, 172)-Net in Base 27 — Constructive

(19, 42, 172)-net in base 27, using

6 times m-reduction [i] based on (19, 48, 172)-net in base 27, using
- base change [i] based on digital (7, 36, 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]

(19, 42, 199)-Net over F₂₇ — Digital

Digital (19, 42, 199)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(19, 42, 226)-Net in Base 27

(19, 42, 226)-net in base 27, using

2 times m-reduction [i] based on (19, 44, 226)-net in base 27, using
- base change [i] based on digital (8, 33, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(19, 42, 40838)-Net in Base 27 — Upper bound on s

There is no (19, 42, 40839)-net in base 27, because

1 times m-reduction [i] would yield (19, 41, 40839)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 48530 136837 835289 367257 782163 407967 465100 479990 756569 287875 > 27⁴¹ [i]