MinT - Best Known (52, 52+53, s)-Nets in Base 27

Best Known (52, 52+53, s)-Nets in Base 27

(52, 52+53, 192)-Net over F₂₇ — Constructive and digital

Digital (52, 105, 192)-net over F₂₇, using

t-expansion [i] based on digital (51, 105, 192)-net over F₂₇, using
- 4 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)

(52, 52+53, 370)-Net in Base 27 — Constructive

(52, 105, 370)-net in base 27, using

t-expansion [i] based on (43, 105, 370)-net in base 27, using
- 3 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]

(52, 52+53, 611)-Net over F₂₇ — Digital

Digital (52, 105, 611)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+53, 215649)-Net in Base 27 — Upper bound on s

There is no (52, 105, 215650)-net in base 27, because

1 times m-reduction [i] would yield (52, 104, 215650)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72758 022498 158255 675177 478331 642527 271684 478947 401229 212174 449091 497731 395396 699082 734239 207366 016045 337801 880391 579342 875691 835030 914963 279974 457477 > 27¹⁰⁴ [i]