MinT - Best Known (87−43, 87, s)-Nets in Base 27

Best Known (87−43, 87, s)-Nets in Base 27

(87−43, 87, 192)-Net over F₂₇ — Constructive and digital

Digital (44, 87, 192)-net over F₂₇, using

1 times m-reduction [i] based on digital (44, 88, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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, 55, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(87−43, 87, 370)-Net in Base 27 — Constructive

(44, 87, 370)-net in base 27, using

t-expansion [i] based on (43, 87, 370)-net in base 27, using
- 21 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]

(87−43, 87, 606)-Net over F₂₇ — Digital

Digital (44, 87, 606)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−43, 87, 242814)-Net in Base 27 — Upper bound on s

There is no (44, 87, 242815)-net in base 27, because

1 times m-reduction [i] would yield (44, 86, 242815)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1251 088617 904042 052956 234193 355623 251995 539751 369146 628361 155784 858850 208193 539622 273814 358063 070979 557973 686012 843739 311839 > 27⁸⁶ [i]