MinT - Best Known (226−83, 226, s)-Nets in Base 3

Best Known (226−83, 226, s)-Nets in Base 3

(226−83, 226, 156)-Net over F₃ — Constructive and digital

Digital (143, 226, 156)-net over F₃, using

16 times m-reduction [i] based on digital (143, 242, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 121, 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]

(226−83, 226, 266)-Net over F₃ — Digital

Digital (143, 226, 266)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−83, 226, 3311)-Net in Base 3 — Upper bound on s

There is no (143, 226, 3312)-net in base 3, because

1 times m-reduction [i] would yield (143, 225, 3312)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 227427 661816 529813 740483 456671 482583 207730 632384 145947 945428 014167 336724 511505 957596 300834 815740 599525 519585 > 3²²⁵ [i]