MinT - Best Known (42, 42+47, s)-Nets in Base 27

Best Known (42, 42+47, s)-Nets in Base 27

(42, 42+47, 182)-Net over F₂₇ — Constructive and digital

Digital (42, 89, 182)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 32, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
2. digital (10, 57, 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]

(42, 42+47, 370)-Net in Base 27 — Constructive

(42, 89, 370)-net in base 27, using

15 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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]

(42, 42+47, 408)-Net over F₂₇ — Digital

Digital (42, 89, 408)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+47, 108631)-Net in Base 27 — Upper bound on s

There is no (42, 89, 108632)-net in base 27, because

1 times m-reduction [i] would yield (42, 88, 108632)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912040 485529 116178 281715 963860 121279 177239 093580 629833 896760 587752 558701 329938 690763 827853 970361 798686 362347 266931 479648 077665 > 27⁸⁸ [i]