MinT - Best Known (49, 49+53, s)-Nets in Base 27

Best Known (49, 49+53, s)-Nets in Base 27

(49, 49+53, 192)-Net over F₂₇ — Constructive and digital

Digital (49, 102, 192)-net over F₂₇, using

1 times m-reduction [i] based on digital (49, 103, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 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, 65, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(49, 49+53, 370)-Net in Base 27 — Constructive

(49, 102, 370)-net in base 27, using

t-expansion [i] based on (43, 102, 370)-net in base 27, using
- 6 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]

(49, 49+53, 500)-Net over F₂₇ — Digital

Digital (49, 102, 500)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+53, 147427)-Net in Base 27 — Upper bound on s

There is no (49, 102, 147428)-net in base 27, because

1 times m-reduction [i] would yield (49, 101, 147428)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696589 734390 237496 010200 769456 711814 804822 827788 530265 300673 640773 760195 210040 254507 825669 666031 719488 639002 681996 881199 902053 872430 874611 016441 > 27¹⁰¹ [i]