MinT - Best Known (186−83, 186, s)-Nets in Base 3

Best Known (186−83, 186, s)-Nets in Base 3

(186−83, 186, 80)-Net over F₃ — Constructive and digital

Digital (103, 186, 80)-net over F₃, using

4 times m-reduction [i] based on digital (103, 190, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 95, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(186−83, 186, 129)-Net over F₃ — Digital

Digital (103, 186, 129)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−83, 186, 1107)-Net in Base 3 — Upper bound on s

There is no (103, 186, 1108)-net in base 3, because

1 times m-reduction [i] would yield (103, 185, 1108)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18797 486740 084677 405303 129529 413222 000916 410363 602434 500666 745598 587521 777263 562186 173097 > 3¹⁸⁵ [i]