MinT - Best Known (123−45, 123, s)-Nets in Base 3

Best Known (123−45, 123, s)-Nets in Base 3

(123−45, 123, 128)-Net over F₃ — Constructive and digital

Digital (78, 123, 128)-net over F₃, using

7 times m-reduction [i] based on digital (78, 130, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 65, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(123−45, 123, 160)-Net over F₃ — Digital

Digital (78, 123, 160)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−45, 123, 1981)-Net in Base 3 — Upper bound on s

There is no (78, 123, 1982)-net in base 3, because

1 times m-reduction [i] would yield (78, 122, 1982)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16248 053421 540045 756451 247764 002807 170482 641759 732882 699717 > 3¹²² [i]