MinT - Best Known (109, 109+89, s)-Nets in Base 3

Best Known (109, 109+89, s)-Nets in Base 3

(109, 109+89, 80)-Net over F₃ — Constructive and digital

Digital (109, 198, 80)-net over F₃, using

4 times m-reduction [i] based on digital (109, 202, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 101, 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]

(109, 109+89, 134)-Net over F₃ — Digital

Digital (109, 198, 134)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+89, 1137)-Net in Base 3 — Upper bound on s

There is no (109, 198, 1138)-net in base 3, because

1 times m-reduction [i] would yield (109, 197, 1138)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9935 777694 260513 954140 842981 203437 463438 290840 146656 040037 962631 659211 162198 584452 717183 326137 > 3¹⁹⁷ [i]