MinT - Best Known (180−69, 180, s)-Nets in Base 3

Best Known (180−69, 180, s)-Nets in Base 3

(180−69, 180, 148)-Net over F₃ — Constructive and digital

Digital (111, 180, 148)-net over F₃, using

8 times m-reduction [i] based on digital (111, 188, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 94, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(180−69, 180, 193)-Net over F₃ — Digital

Digital (111, 180, 193)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−69, 180, 2166)-Net in Base 3 — Upper bound on s

There is no (111, 180, 2167)-net in base 3, because

1 times m-reduction [i] would yield (111, 179, 2167)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 654536 556312 229243 655405 719093 695334 707196 629076 006198 478432 579060 452305 676047 776045 > 3¹⁷⁹ [i]