MinT - Best Known (246−83, 246, s)-Nets in Base 3

Best Known (246−83, 246, s)-Nets in Base 3

(246−83, 246, 162)-Net over F₃ — Constructive and digital

Digital (163, 246, 162)-net over F₃, using

t-expansion [i] based on digital (157, 246, 162)-net over F₃, using
- 4 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]

(246−83, 246, 367)-Net over F₃ — Digital

Digital (163, 246, 367)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−83, 246, 5687)-Net in Base 3 — Upper bound on s

There is no (163, 246, 5688)-net in base 3, because

1 times m-reduction [i] would yield (163, 245, 5688)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 789 056892 727564 578735 268212 335010 453441 176687 731666 405160 689102 348310 294643 560158 645717 720012 356509 789916 396333 083889 > 3²⁴⁵ [i]