MinT - Best Known (212−72, 212, s)-Nets in Base 3

Best Known (212−72, 212, s)-Nets in Base 3

(212−72, 212, 162)-Net over F₃ — Constructive and digital

Digital (140, 212, 162)-net over F₃, using

4 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]

(212−72, 212, 316)-Net over F₃ — Digital

Digital (140, 212, 316)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−72, 212, 4571)-Net in Base 3 — Upper bound on s

There is no (140, 212, 4572)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141550 379472 773578 143279 309690 569243 669915 991111 504545 020529 747023 735292 379154 485750 478796 043338 407425 > 3²¹² [i]