MinT - Best Known (50, 50+59, s)-Nets in Base 27

Best Known (50, 50+59, s)-Nets in Base 27

(50, 50+59, 190)-Net over F₂₇ — Constructive and digital

Digital (50, 109, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 39, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (11, 70, 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]

(50, 50+59, 370)-Net in Base 27 — Constructive

(50, 109, 370)-net in base 27, using

27¹ times duplication [i] based on (49, 108, 370)-net in base 27, using
- t-expansion [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+59, 414)-Net over F₂₇ — Digital

Digital (50, 109, 414)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+59, 96089)-Net in Base 27 — Upper bound on s

There is no (50, 109, 96090)-net in base 27, because

1 times m-reduction [i] would yield (50, 108, 96090)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38666 675223 060580 301728 024515 582782 912676 177884 490114 329590 301988 083755 830507 123341 183558 554111 860224 403702 775090 272088 543256 959089 263041 275154 872761 837325 > 27¹⁰⁸ [i]