MinT - Best Known (111, 111+89, s)-Nets in Base 3

Best Known (111, 111+89, s)-Nets in Base 3

(111, 111+89, 80)-Net over F₃ — Constructive and digital

Digital (111, 200, 80)-net over F₃, using

6 times m-reduction [i] based on digital (111, 206, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 103, 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]

(111, 111+89, 139)-Net over F₃ — Digital

Digital (111, 200, 139)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+89, 1198)-Net in Base 3 — Upper bound on s

There is no (111, 200, 1199)-net in base 3, because

1 times m-reduction [i] would yield (111, 199, 1199)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 91235 973124 955927 184061 362433 268388 972562 702109 228617 155672 871129 185837 342073 712264 408802 286761 > 3¹⁹⁹ [i]