MinT - Best Known (249−87, 249, s)-Nets in Base 3

Best Known (249−87, 249, s)-Nets in Base 3

(249−87, 249, 162)-Net over F₃ — Constructive and digital

Digital (162, 249, 162)-net over F₃, using

t-expansion [i] based on digital (157, 249, 162)-net over F₃, using
- 1 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 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]

(249−87, 249, 334)-Net over F₃ — Digital

Digital (162, 249, 334)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−87, 249, 4724)-Net in Base 3 — Upper bound on s

There is no (162, 249, 4725)-net in base 3, because

1 times m-reduction [i] would yield (162, 248, 4725)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21331 762566 664428 789196 244000 331816 751925 868081 766133 194389 626046 708249 868143 255319 788468 762031 814453 268979 300229 092299 > 3²⁴⁸ [i]