MinT - Best Known (43, 43+29, s)-Nets in Base 27

Best Known (43, 43+29, s)-Nets in Base 27

(43, 43+29, 240)-Net over F₂₇ — Constructive and digital

Digital (43, 72, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 15, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 21, 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]
3. digital (7, 36, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(43, 43+29, 370)-Net in Base 27 — Constructive

(43, 72, 370)-net in base 27, using

36 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 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]

(43, 43+29, 2097)-Net over F₂₇ — Digital

Digital (43, 72, 2097)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(43, 43+29, 4222238)-Net in Base 27 — Upper bound on s

There is no (43, 72, 4222239)-net in base 27, because

1 times m-reduction [i] would yield (43, 71, 4222239)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 423475 252833 680435 430015 519078 560811 796140 485944 404966 246297 963778 640228 552590 444689 302273 779381 402581 > 27⁷¹ [i]