MinT - Best Known (51, 104, s)-Nets in Base 27

Best Known (51, 104, s)-Nets in Base 27

(51, 104, 192)-Net over F₂₇ — Constructive and digital

Digital (51, 104, 192)-net over F₂₇, using

5 times m-reduction [i] based on digital (51, 109, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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, 69, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(51, 104, 370)-Net in Base 27 — Constructive

(51, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 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]

(51, 104, 571)-Net over F₂₇ — Digital

Digital (51, 104, 571)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 104, 189972)-Net in Base 27 — Upper bound on s

There is no (51, 104, 189973)-net in base 27, because

1 times m-reduction [i] would yield (51, 103, 189973)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 506911 630941 720401 961485 414708 036170 864494 598937 731552 392101 156531 308743 842606 525861 446542 603125 112383 038657 291497 077881 506051 268454 145084 787425 > 27¹⁰³ [i]