MinT - Best Known (247−89, 247, s)-Nets in Base 3

Best Known (247−89, 247, s)-Nets in Base 3

(247−89, 247, 162)-Net over F₃ — Constructive and digital

Digital (158, 247, 162)-net over F₃, using

t-expansion [i] based on digital (157, 247, 162)-net over F₃, using
- 3 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]

(247−89, 247, 304)-Net over F₃ — Digital

Digital (158, 247, 304)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−89, 247, 3969)-Net in Base 3 — Upper bound on s

There is no (158, 247, 3970)-net in base 3, because

1 times m-reduction [i] would yield (158, 246, 3970)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2370 350935 250312 990848 307299 809265 323387 226017 817528 903514 372516 630336 081519 433259 175331 000303 477124 056265 647627 381945 > 3²⁴⁶ [i]