MinT - Best Known (212−77, 212, s)-Nets in Base 3

Best Known (212−77, 212, s)-Nets in Base 3

(212−77, 212, 156)-Net over F₃ — Constructive and digital

Digital (135, 212, 156)-net over F₃, using

14 times m-reduction [i] based on digital (135, 226, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 113, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(212−77, 212, 259)-Net over F₃ — Digital

Digital (135, 212, 259)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−77, 212, 3312)-Net in Base 3 — Upper bound on s

There is no (135, 212, 3313)-net in base 3, because

1 times m-reduction [i] would yield (135, 211, 3313)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47198 362715 299098 915424 894724 870169 121326 512570 287904 794066 555850 999652 050444 558673 458171 684526 129849 > 3²¹¹ [i]