MinT - Best Known (45, 45+47, s)-Nets in Base 27

Best Known (45, 45+47, s)-Nets in Base 27

(45, 45+47, 192)-Net over F₂₇ — Constructive and digital

Digital (45, 92, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 34, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]
2. digital (11, 58, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(45, 45+47, 370)-Net in Base 27 — Constructive

(45, 92, 370)-net in base 27, using

t-expansion [i] based on (43, 92, 370)-net in base 27, using
- 16 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]

(45, 45+47, 512)-Net over F₂₇ — Digital

Digital (45, 92, 512)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+47, 166983)-Net in Base 27 — Upper bound on s

There is no (45, 92, 166984)-net in base 27, because

1 times m-reduction [i] would yield (45, 91, 166984)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17952 512723 127602 360015 595873 365654 323270 301418 301966 772492 455407 878832 166802 158045 571411 029191 121490 787806 840201 479108 670283 653409 > 27⁹¹ [i]