MinT - Best Known (54−39, 54, s)-Nets in Base 27

Best Known (54−39, 54, s)-Nets in Base 27

(54−39, 54, 96)-Net over F₂₇ — Constructive and digital

Digital (15, 54, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 54, 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]

(54−39, 54, 100)-Net in Base 27 — Constructive

(15, 54, 100)-net in base 27, using

2 times m-reduction [i] based on (15, 56, 100)-net in base 27, using
- base change [i] based on digital (1, 42, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(54−39, 54, 136)-Net over F₂₇ — Digital

Digital (15, 54, 136)-net over F₂₇, using

t-expansion [i] based on digital (13, 54, 136)-net over F₂₇, using
- net from sequence [i] based on digital (13, 135)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 13 and N(F) ≥ 136, using
    - a curve from the manYPoints database [i]

(54−39, 54, 2989)-Net in Base 27 — Upper bound on s

There is no (15, 54, 2990)-net in base 27, because

1 times m-reduction [i] would yield (15, 53, 2990)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 7310 418897 165180 681802 198012 538914 106610 735740 579312 055845 912521 906267 453553 > 27⁵³ [i]