MinT - Best Known (50, 50+53, s)-Nets in Base 27

Best Known (50, 50+53, s)-Nets in Base 27

(50, 50+53, 192)-Net over F₂₇ — Constructive and digital

Digital (50, 103, 192)-net over F₂₇, using

3 times m-reduction [i] based on digital (50, 106, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 39, 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, 67, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(50, 50+53, 370)-Net in Base 27 — Constructive

(50, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 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]

(50, 50+53, 535)-Net over F₂₇ — Digital

Digital (50, 103, 535)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+53, 167353)-Net in Base 27 — Upper bound on s

There is no (50, 103, 167354)-net in base 27, because

1 times m-reduction [i] would yield (50, 102, 167354)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 801566 685360 503516 745600 884739 478727 599428 904572 016918 872808 114110 900813 613725 664139 385752 333229 906660 844143 159530 222081 826535 702419 837819 374805 > 27¹⁰² [i]