MinT - Best Known (245−97, 245, s)-Nets in Base 3

Best Known (245−97, 245, s)-Nets in Base 3

(245−97, 245, 156)-Net over F₃ — Constructive and digital

Digital (148, 245, 156)-net over F₃, using

t-expansion [i] based on digital (147, 245, 156)-net over F₃, using
- 5 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
  - trace code for nets [i] based on digital (22, 125, 78)-net over F₉, using
    - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
        F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(245−97, 245, 230)-Net over F₃ — Digital

Digital (148, 245, 230)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−97, 245, 2448)-Net in Base 3 — Upper bound on s

There is no (148, 245, 2449)-net in base 3, because

1 times m-reduction [i] would yield (148, 244, 2449)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 265 661853 648499 609645 314422 532627 195894 605965 734287 300008 463196 526714 328822 100712 755800 060143 212573 844093 477693 659585 > 3²⁴⁴ [i]