MinT - Best Known (220−87, 220, s)-Nets in Base 3

Best Known (220−87, 220, s)-Nets in Base 3

(220−87, 220, 156)-Net over F₃ — Constructive and digital

Digital (133, 220, 156)-net over F₃, using

2 times m-reduction [i] based on digital (133, 222, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 111, 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]

(220−87, 220, 210)-Net over F₃ — Digital

Digital (133, 220, 210)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−87, 220, 2229)-Net in Base 3 — Upper bound on s

There is no (133, 220, 2230)-net in base 3, because

1 times m-reduction [i] would yield (133, 219, 2230)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 309 108796 655565 434970 009481 269951 613070 105735 853988 998932 067342 648529 043536 934430 775105 255175 987470 163217 > 3²¹⁹ [i]