MinT - Best Known (249−97, 249, s)-Nets in Base 3

Best Known (249−97, 249, s)-Nets in Base 3

(249−97, 249, 156)-Net over F₃ — Constructive and digital

Digital (152, 249, 156)-net over F₃, using

t-expansion [i] based on digital (147, 249, 156)-net over F₃, using
- 1 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]

(249−97, 249, 244)-Net over F₃ — Digital

Digital (152, 249, 244)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−97, 249, 2687)-Net in Base 3 — Upper bound on s

There is no (152, 249, 2688)-net in base 3, because

1 times m-reduction [i] would yield (152, 248, 2688)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21413 856706 126424 249058 758176 595446 582588 482664 677326 907384 570084 382120 492280 903891 478751 192892 404069 872496 806264 676353 > 3²⁴⁸ [i]