MinT - Best Known (52−35, 52, s)-Nets in Base 27

Best Known (52−35, 52, s)-Nets in Base 27

(52−35, 52, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 52, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 52, 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]

(52−35, 52, 144)-Net over F₂₇ — Digital

Digital (17, 52, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 52, 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]

(52−35, 52, 150)-Net in Base 27 — Constructive

(17, 52, 150)-net in base 27, using

base change [i] based on digital (4, 39, 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]

(52−35, 52, 154)-Net in Base 27

(17, 52, 154)-net in base 27, using

base change [i] based on digital (4, 39, 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]

(52−35, 52, 5424)-Net in Base 27 — Upper bound on s

There is no (17, 52, 5425)-net in base 27, because

1 times m-reduction [i] would yield (17, 51, 5425)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 10 000863 474142 637147 939660 618061 368794 726175 310461 113106 777780 815540 882491 > 27⁵¹ [i]