MinT - Best Known (41, 41+49, s)-Nets in Base 27

Best Known (41, 41+49, s)-Nets in Base 27

(41, 41+49, 176)-Net over F₂₇ — Constructive and digital

Digital (41, 90, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 31, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (10, 59, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(41, 41+49, 344)-Net over F₂₇ — Digital

Digital (41, 90, 344)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+49, 370)-Net in Base 27 — Constructive

(41, 90, 370)-net in base 27, using

10 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(41, 41+49, 76611)-Net in Base 27 — Upper bound on s

There is no (41, 90, 76612)-net in base 27, because

1 times m-reduction [i] would yield (41, 89, 76612)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 630194 549719 615667 330624 776019 296035 750965 246842 911538 116399 712350 612484 164638 134420 476611 655805 783097 141412 022454 565964 122177 > 27⁸⁹ [i]