MinT - Best Known (190−61, 190, s)-Nets in Base 3

Best Known (190−61, 190, s)-Nets in Base 3

(190−61, 190, 162)-Net over F₃ — Constructive and digital

Digital (129, 190, 162)-net over F₃, using

4 times m-reduction [i] based on digital (129, 194, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 97, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(190−61, 190, 344)-Net over F₃ — Digital

Digital (129, 190, 344)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−61, 190, 6074)-Net in Base 3 — Upper bound on s

There is no (129, 190, 6075)-net in base 3, because

1 times m-reduction [i] would yield (129, 189, 6075)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 501813 209947 354744 796882 717230 477130 075916 895668 238594 938917 886299 263344 106737 643754 294877 > 3¹⁸⁹ [i]