MinT - Best Known (50, 50+49, s)-Nets in Base 27

Best Known (50, 50+49, s)-Nets in Base 27

(50, 50+49, 192)-Net over F₂₇ — Constructive and digital

Digital (50, 99, 192)-net over F₂₇, using

7 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+49, 370)-Net in Base 27 — Constructive

(50, 99, 370)-net in base 27, using

t-expansion [i] based on (43, 99, 370)-net in base 27, using
- 9 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+49, 661)-Net over F₂₇ — Digital

Digital (50, 99, 661)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+49, 263696)-Net in Base 27 — Upper bound on s

There is no (50, 99, 263697)-net in base 27, because

1 times m-reduction [i] would yield (50, 98, 263697)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 789040 531707 856143 890849 442433 039330 933013 653772 260812 897732 851414 389265 721060 378911 953975 254372 047196 961144 138381 361323 282922 060240 036513 > 27⁹⁸ [i]