MinT - Best Known (48−31, 48, s)-Nets in Base 27

Best Known (48−31, 48, s)-Nets in Base 27

(48−31, 48, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 48, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 48, 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]

(48−31, 48, 144)-Net over F₂₇ — Digital

Digital (17, 48, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 48, 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]

(48−31, 48, 160)-Net in Base 27 — Constructive

(17, 48, 160)-net in base 27, using

base change [i] based on digital (5, 36, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(48−31, 48, 167)-Net in Base 27

(17, 48, 167)-net in base 27, using

base change [i] based on digital (5, 36, 167)-net over F₈₁, using
- net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
    - a curve from the manYPoints database [i]

(48−31, 48, 7538)-Net in Base 27 — Upper bound on s

There is no (17, 48, 7539)-net in base 27, because

1 times m-reduction [i] would yield (17, 47, 7539)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 18 808281 216938 408983 963201 535305 995383 787971 180136 996170 755885 855083 > 27⁴⁷ [i]