MinT - Best Known (69−43, 69, s)-Nets in Base 27

Best Known (69−43, 69, s)-Nets in Base 27

(69−43, 69, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 69, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 69, 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]

(69−43, 69, 172)-Net in Base 27 — Constructive

(26, 69, 172)-net in base 27, using

7 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]

(69−43, 69, 208)-Net over F₂₇ — Digital

Digital (26, 69, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 69, 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]

(69−43, 69, 226)-Net in Base 27

(26, 69, 226)-net in base 27, using

3 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]

(69−43, 69, 14390)-Net in Base 27 — Upper bound on s

There is no (26, 69, 14391)-net in base 27, because

1 times m-reduction [i] would yield (26, 68, 14391)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 520501 025073 916785 025224 695710 394080 168968 447980 446922 370312 263729 098618 331708 075335 783638 489679 > 27⁶⁸ [i]