MinT - Best Known (243−87, 243, s)-Nets in Base 3

Best Known (243−87, 243, s)-Nets in Base 3

(243−87, 243, 162)-Net over F₃ — Constructive and digital

Digital (156, 243, 162)-net over F₃, using

5 times m-reduction [i] based on digital (156, 248, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 124, 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]

(243−87, 243, 305)-Net over F₃ — Digital

Digital (156, 243, 305)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−87, 243, 4046)-Net in Base 3 — Upper bound on s

There is no (156, 243, 4047)-net in base 3, because

1 times m-reduction [i] would yield (156, 242, 4047)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 115550 397619 073476 163566 835478 935599 269235 477721 239445 506168 517992 528468 888261 548152 317518 971014 561384 515517 639915 > 3²⁴² [i]