MinT - Best Known (81−55, 81, s)-Nets in Base 27

Best Known (81−55, 81, s)-Nets in Base 27

(81−55, 81, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 81, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 81, 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]

(81−55, 81, 160)-Net in Base 27 — Constructive

(26, 81, 160)-net in base 27, using

3 times m-reduction [i] based on (26, 84, 160)-net in base 27, using
- base change [i] based on digital (5, 63, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(81−55, 81, 208)-Net over F₂₇ — Digital

Digital (26, 81, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 81, 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]

(81−55, 81, 7305)-Net in Base 27 — Upper bound on s

There is no (26, 81, 7306)-net in base 27, because

1 times m-reduction [i] would yield (26, 80, 7306)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 229599 745460 846245 244931 164555 281239 397127 361303 773044 587644 038610 619367 148737 266047 252236 209807 433493 247950 039601 > 27⁸⁰ [i]