MinT - Best Known (218−83, 218, s)-Nets in Base 3

Best Known (218−83, 218, s)-Nets in Base 3

(218−83, 218, 156)-Net over F₃ — Constructive and digital

Digital (135, 218, 156)-net over F₃, using

8 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]

(218−83, 218, 232)-Net over F₃ — Digital

Digital (135, 218, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−83, 218, 2664)-Net in Base 3 — Upper bound on s

There is no (135, 218, 2665)-net in base 3, because

1 times m-reduction [i] would yield (135, 217, 2665)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 538030 275164 965950 794414 152450 272730 690082 649992 906902 496741 588521 946650 436665 247082 049649 965304 111795 > 3²¹⁷ [i]