MinT - Best Known (16, 16+39, s)-Nets in Base 27

Best Known (16, 16+39, s)-Nets in Base 27

(16, 16+39, 96)-Net over F₂₇ — Constructive and digital

Digital (16, 55, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 55, 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]

(16, 16+39, 116)-Net in Base 27 — Constructive

(16, 55, 116)-net in base 27, using

1 times m-reduction [i] based on (16, 56, 116)-net in base 27, using
- base change [i] based on digital (2, 42, 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]

(16, 16+39, 144)-Net over F₂₇ — Digital

Digital (16, 55, 144)-net over F₂₇, using

net from sequence [i] based on digital (16, 143)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 16 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(16, 16+39, 3557)-Net in Base 27 — Upper bound on s

There is no (16, 55, 3558)-net in base 27, because

1 times m-reduction [i] would yield (16, 54, 3558)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 197167 009768 216619 348115 226877 300247 282107 706854 079855 058034 988500 868943 099345 > 27⁵⁴ [i]