MinT - Best Known (16, 16+41, s)-Nets in Base 27

Best Known (16, 16+41, s)-Nets in Base 27

(16, 16+41, 96)-Net over F₂₇ — Constructive and digital

Digital (16, 57, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 57, 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+41, 100)-Net in Base 27 — Constructive

(16, 57, 100)-net in base 27, using

3 times m-reduction [i] based on (16, 60, 100)-net in base 27, using
- base change [i] based on digital (1, 45, 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]

(16, 16+41, 144)-Net over F₂₇ — Digital

Digital (16, 57, 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+41, 3241)-Net in Base 27 — Upper bound on s

There is no (16, 57, 3242)-net in base 27, because

1 times m-reduction [i] would yield (16, 56, 3242)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 143 566224 058144 136388 653452 876557 974471 803711 156690 958638 334043 514537 939679 972937 > 27⁵⁶ [i]