MinT - Best Known (58−35, 58, s)-Nets in Base 27

Best Known (58−35, 58, s)-Nets in Base 27

(58−35, 58, 114)-Net over F₂₇ — Constructive and digital

Digital (23, 58, 114)-net over F₂₇, using

net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
  - an explicitly constructive algebraic function field [i]

(58−35, 58, 163)-Net over F₂₇ — Digital

Digital (23, 58, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 58, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(58−35, 58, 172)-Net in Base 27 — Constructive

(23, 58, 172)-net in base 27, using

6 times m-reduction [i] based on (23, 64, 172)-net in base 27, using
- base change [i] based on digital (7, 48, 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]

(58−35, 58, 226)-Net in Base 27

(23, 58, 226)-net in base 27, using

2 times m-reduction [i] based on (23, 60, 226)-net in base 27, using
- base change [i] based on digital (8, 45, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(58−35, 58, 17379)-Net in Base 27 — Upper bound on s

There is no (23, 58, 17380)-net in base 27, because

1 times m-reduction [i] would yield (23, 57, 17380)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3872 570481 363693 365614 002527 219850 864623 623184 442606 044741 064410 275196 702185 848873 > 27⁵⁷ [i]