MinT - Best Known (41, 41+43, s)-Nets in Base 27

Best Known (41, 41+43, s)-Nets in Base 27

(41, 41+43, 188)-Net over F₂₇ — Constructive and digital

Digital (41, 84, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 31, 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]
2. digital (10, 53, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(41, 41+43, 370)-Net in Base 27 — Constructive

(41, 84, 370)-net in base 27, using

16 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+43, 473)-Net over F₂₇ — Digital

Digital (41, 84, 473)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+43, 151629)-Net in Base 27 — Upper bound on s

There is no (41, 84, 151630)-net in base 27, because

1 times m-reduction [i] would yield (41, 83, 151630)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63567 371450 365335 932686 084797 332938 703022 392676 729257 687549 646834 849398 950272 780705 282833 301992 709671 922345 163175 987941 > 27⁸³ [i]