MinT - Best Known (246−99, 246, s)-Nets in Base 3

Best Known (246−99, 246, s)-Nets in Base 3

(246−99, 246, 156)-Net over F₃ — Constructive and digital

Digital (147, 246, 156)-net over F₃, using

4 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]

(246−99, 246, 220)-Net over F₃ — Digital

Digital (147, 246, 220)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−99, 246, 2274)-Net in Base 3 — Upper bound on s

There is no (147, 246, 2275)-net in base 3, because

1 times m-reduction [i] would yield (147, 245, 2275)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 800 811207 472635 542192 296478 705097 127797 704553 092233 045087 556756 567920 357823 980225 398614 282160 562509 943713 049832 313063 > 3²⁴⁵ [i]