MinT - Best Known (26, 26+45, s)-Nets in Base 27

Best Known (26, 26+45, s)-Nets in Base 27

(26, 26+45, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 71, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 71, 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]

(26, 26+45, 172)-Net in Base 27 — Constructive

(26, 71, 172)-net in base 27, using

5 times m-reduction [i] based on (26, 76, 172)-net in base 27, using
- base change [i] based on digital (7, 57, 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]

(26, 26+45, 208)-Net over F₂₇ — Digital

Digital (26, 71, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 71, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(26, 26+45, 226)-Net in Base 27

(26, 71, 226)-net in base 27, using

1 times m-reduction [i] based on (26, 72, 226)-net in base 27, using
- base change [i] based on digital (8, 54, 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]

(26, 26+45, 12468)-Net in Base 27 — Upper bound on s

There is no (26, 71, 12469)-net in base 27, because

1 times m-reduction [i] would yield (26, 70, 12469)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 15690 296347 506705 063903 604613 891558 129689 447400 911461 872691 769918 101894 124772 701825 023705 901764 277713 > 27⁷⁰ [i]