MinT - Best Known (50−31, 50, s)-Nets in Base 27

Best Known (50−31, 50, s)-Nets in Base 27

(50−31, 50, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 50, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 50, 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]

(50−31, 50, 148)-Net over F₂₇ — Digital

Digital (19, 50, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 50, 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]

(50−31, 50, 160)-Net in Base 27 — Constructive

(19, 50, 160)-net in base 27, using

6 times m-reduction [i] based on (19, 56, 160)-net in base 27, using
- base change [i] based on digital (5, 42, 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]

(50−31, 50, 190)-Net in Base 27

(19, 50, 190)-net in base 27, using

2 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]

(50−31, 50, 11703)-Net in Base 27 — Upper bound on s

There is no (19, 50, 11704)-net in base 27, because

1 times m-reduction [i] would yield (19, 49, 11704)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13719 257097 289758 114727 059188 853714 995009 213084 398299 157972 850580 527073 > 27⁴⁹ [i]