MinT - Best Known (84−47, 84, s)-Nets in Base 27

Best Known (84−47, 84, s)-Nets in Base 27

(84−47, 84, 164)-Net over F₂₇ — Constructive and digital

Digital (37, 84, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 30, 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 (7, 54, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(84−47, 84, 279)-Net over F₂₇ — Digital

Digital (37, 84, 279)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−47, 84, 370)-Net in Base 27 — Constructive

(37, 84, 370)-net in base 27, using

base change [i] based on digital (16, 63, 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]

(84−47, 84, 53056)-Net in Base 27 — Upper bound on s

There is no (37, 84, 53057)-net in base 27, because

1 times m-reduction [i] would yield (37, 83, 53057)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 63564 046677 252530 776309 439326 183806 039232 467339 217359 294693 949371 927727 788174 918865 138142 224486 293077 742210 951407 580915 > 27⁸³ [i]