MinT - Best Known (67−42, 67, s)-Nets in Base 27

Best Known (67−42, 67, s)-Nets in Base 27

(67−42, 67, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 67, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 67, 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]

(67−42, 67, 172)-Net in Base 27 — Constructive

(25, 67, 172)-net in base 27, using

5 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 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]

(67−42, 67, 208)-Net over F₂₇ — Digital

Digital (25, 67, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 67, 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]

(67−42, 67, 226)-Net in Base 27

(25, 67, 226)-net in base 27, using

1 times m-reduction [i] based on (25, 68, 226)-net in base 27, using
- base change [i] based on digital (8, 51, 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]

(67−42, 67, 12299)-Net in Base 27 — Upper bound on s

There is no (25, 67, 12300)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 798201 022681 255252 609899 269560 214397 289878 863897 227089 188556 554300 909026 788506 759868 926003 676441 > 27⁶⁷ [i]