MinT - Best Known (229−91, 229, s)-Nets in Base 3

Best Known (229−91, 229, s)-Nets in Base 3

(229−91, 229, 156)-Net over F₃ — Constructive and digital

Digital (138, 229, 156)-net over F₃, using

3 times m-reduction [i] based on digital (138, 232, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 116, 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]

(229−91, 229, 214)-Net over F₃ — Digital

Digital (138, 229, 214)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−91, 229, 2260)-Net in Base 3 — Upper bound on s

There is no (138, 229, 2261)-net in base 3, because

1 times m-reduction [i] would yield (138, 228, 2261)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 165398 931515 501453 933796 544253 646663 716677 796660 584771 739972 646715 808489 856172 629466 527073 748336 130762 127387 > 3²²⁸ [i]