MinT - Best Known (19, 64, s)-Nets in Base 27

Best Known (19, 64, s)-Nets in Base 27

(19, 64, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 64, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 64, 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, 64, 116)-Net in Base 27 — Constructive

(19, 64, 116)-net in base 27, using

4 times m-reduction [i] based on (19, 68, 116)-net in base 27, using
- base change [i] based on digital (2, 51, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(19, 64, 148)-Net over F₂₇ — Digital

Digital (19, 64, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 64, 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, 64, 4361)-Net in Base 27 — Upper bound on s

There is no (19, 64, 4362)-net in base 27, because

1 times m-reduction [i] would yield (19, 63, 4362)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 501540 961557 169111 865512 756055 257100 566723 987046 152716 137424 996578 979895 540627 662119 667917 > 27⁶³ [i]