MinT - Best Known (85−59, 85, s)-Nets in Base 27

Best Known (85−59, 85, s)-Nets in Base 27

(85−59, 85, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 85, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 85, 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]

(85−59, 85, 150)-Net in Base 27 — Constructive

(26, 85, 150)-net in base 27, using

3 times m-reduction [i] based on (26, 88, 150)-net in base 27, using
- base change [i] based on digital (4, 66, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(85−59, 85, 208)-Net over F₂₇ — Digital

Digital (26, 85, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 85, 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]

(85−59, 85, 6267)-Net in Base 27 — Upper bound on s

There is no (26, 85, 6268)-net in base 27, because

1 times m-reduction [i] would yield (26, 84, 6268)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716911 708887 425085 457811 248535 463558 978607 479730 610785 789836 222333 819044 299885 991509 647974 823481 530812 597654 493929 051513 > 27⁸⁴ [i]