MinT - Best Known (229−85, 229, s)-Nets in Base 3

Best Known (229−85, 229, s)-Nets in Base 3

(229−85, 229, 156)-Net over F₃ — Constructive and digital

Digital (144, 229, 156)-net over F₃, using

15 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 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]

(229−85, 229, 261)-Net over F₃ — Digital

Digital (144, 229, 261)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−85, 229, 3171)-Net in Base 3 — Upper bound on s

There is no (144, 229, 3172)-net in base 3, because

1 times m-reduction [i] would yield (144, 228, 3172)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 129276 631377 705209 322517 145718 044019 477143 282275 720022 220011 760210 832537 675882 905780 499563 496368 373591 466745 > 3²²⁸ [i]