MinT - Best Known (66−41, 66, s)-Nets in Base 27

Best Known (66−41, 66, s)-Nets in Base 27

(66−41, 66, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 66, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 66, 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]

(66−41, 66, 172)-Net in Base 27 — Constructive

(25, 66, 172)-net in base 27, using

6 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]

(66−41, 66, 208)-Net over F₂₇ — Digital

Digital (25, 66, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 66, 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]

(66−41, 66, 226)-Net in Base 27

(25, 66, 226)-net in base 27, using

2 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]

(66−41, 66, 14320)-Net in Base 27 — Upper bound on s

There is no (25, 66, 14321)-net in base 27, because

1 times m-reduction [i] would yield (25, 65, 14321)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1094 043059 937862 255898 168979 940997 013633 182056 505150 688364 026691 900529 830629 045261 084797 272881 > 27⁶⁵ [i]