MinT - Best Known (220−77, 220, s)-Nets in Base 3

Best Known (220−77, 220, s)-Nets in Base 3

(220−77, 220, 162)-Net over F₃ — Constructive and digital

Digital (143, 220, 162)-net over F₃, using

2 times m-reduction [i] based on digital (143, 222, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 111, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(220−77, 220, 298)-Net over F₃ — Digital

Digital (143, 220, 298)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−77, 220, 4184)-Net in Base 3 — Upper bound on s

There is no (143, 220, 4185)-net in base 3, because

1 times m-reduction [i] would yield (143, 219, 4185)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 310 259737 830233 477918 870455 945905 533118 039099 285636 886846 791589 144105 041720 081638 421848 553056 952323 784137 > 3²¹⁹ [i]