MinT - Best Known (51−31, 51, s)-Nets in Base 27

Best Known (51−31, 51, s)-Nets in Base 27

(51−31, 51, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 51, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 51, 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]

(51−31, 51, 148)-Net over F₂₇ — Digital

Digital (20, 51, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 51, 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]

(51−31, 51, 172)-Net in Base 27 — Constructive

(20, 51, 172)-net in base 27, using

1 times m-reduction [i] based on (20, 52, 172)-net in base 27, using
- base change [i] based on digital (7, 39, 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]

(51−31, 51, 190)-Net in Base 27

(20, 51, 190)-net in base 27, using

5 times m-reduction [i] based on (20, 56, 190)-net in base 27, using
- base change [i] based on digital (6, 42, 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]

(51−31, 51, 14580)-Net in Base 27 — Upper bound on s

There is no (20, 51, 14581)-net in base 27, because

1 times m-reduction [i] would yield (20, 50, 14581)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 370078 742498 360225 563586 625999 257355 259863 609995 625628 525679 830202 110275 > 27⁵⁰ [i]