MinT - Best Known (215−75, 215, s)-Nets in Base 3

Best Known (215−75, 215, s)-Nets in Base 3

(215−75, 215, 162)-Net over F₃ — Constructive and digital

Digital (140, 215, 162)-net over F₃, using

1 times m-reduction [i] based on digital (140, 216, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 108, 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]

(215−75, 215, 295)-Net over F₃ — Digital

Digital (140, 215, 295)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−75, 215, 4175)-Net in Base 3 — Upper bound on s

There is no (140, 215, 4176)-net in base 3, because

1 times m-reduction [i] would yield (140, 214, 4176)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 278934 519426 220263 664167 101295 039167 834419 984445 002262 812041 422832 109322 375893 776099 141622 548656 744097 > 3²¹⁴ [i]