MinT - Best Known (180−57, 180, s)-Nets in Base 3

Best Known (180−57, 180, s)-Nets in Base 3

(180−57, 180, 162)-Net over F₃ — Constructive and digital

Digital (123, 180, 162)-net over F₃, using

2 times m-reduction [i] based on digital (123, 182, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 91, 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]

(180−57, 180, 342)-Net over F₃ — Digital

Digital (123, 180, 342)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−57, 180, 6313)-Net in Base 3 — Upper bound on s

There is no (123, 180, 6314)-net in base 3, because

1 times m-reduction [i] would yield (123, 179, 6314)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 467106 681007 348284 574907 192694 015972 162917 999403 278002 802474 223634 948064 556446 956825 > 3¹⁷⁹ [i]