MinT - Best Known (19, 19+29, s)-Nets in Base 27

Best Known (19, 19+29, s)-Nets in Base 27

(19, 19+29, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 48, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 48, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(19, 19+29, 148)-Net over F₂₇ — Digital

Digital (19, 48, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 48, 148)-net over F₂₇, using
- net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
    - a curve from the manYPoints database [i]

(19, 19+29, 172)-Net in Base 27 — Constructive

(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, 19+29, 190)-Net in Base 27

(19, 48, 190)-net in base 27, using

4 times m-reduction [i] based on (19, 52, 190)-net in base 27, using
- base change [i] based on digital (6, 39, 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]

(19, 19+29, 14844)-Net in Base 27 — Upper bound on s

There is no (19, 48, 14845)-net in base 27, because

1 times m-reduction [i] would yield (19, 47, 14845)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 18 802825 195929 752012 296646 200002 600716 679444 476148 055463 193733 059009 > 27⁴⁷ [i]