MinT - Best Known (81−53, 81, s)-Nets in Base 27

Best Known (81−53, 81, s)-Nets in Base 27

(81−53, 81, 114)-Net over F₂₇ — Constructive and digital

Digital (28, 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]

(81−53, 81, 172)-Net in Base 27 — Constructive

(28, 81, 172)-net in base 27, using

3 times m-reduction [i] based on (28, 84, 172)-net in base 27, using
- base change [i] based on digital (7, 63, 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]

(81−53, 81, 208)-Net over F₂₇ — Digital

Digital (28, 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]

(81−53, 81, 10278)-Net in Base 27 — Upper bound on s

There is no (28, 81, 10279)-net in base 27, because

1 times m-reduction [i] would yield (28, 80, 10279)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 230177 647531 380386 459603 376400 828026 452790 918467 890487 510939 065015 712055 671592 945841 107101 554395 143227 185135 505965 > 27⁸⁰ [i]