MinT - Best Known (222−77, 222, s)-Nets in Base 3

Best Known (222−77, 222, s)-Nets in Base 3

(222−77, 222, 162)-Net over F₃ — Constructive and digital

Digital (145, 222, 162)-net over F₃, using

4 times m-reduction [i] based on digital (145, 226, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 113, 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]

(222−77, 222, 309)-Net over F₃ — Digital

Digital (145, 222, 309)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−77, 222, 4435)-Net in Base 3 — Upper bound on s

There is no (145, 222, 4436)-net in base 3, because

1 times m-reduction [i] would yield (145, 221, 4436)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2784 454074 696485 910633 905060 880742 738011 244043 845215 402361 017215 108154 606686 701341 599553 823702 084367 024313 > 3²²¹ [i]