MinT - Best Known (123−51, 123, s)-Nets in Base 3

Best Known (123−51, 123, s)-Nets in Base 3

(123−51, 123, 80)-Net over F₃ — Constructive and digital

Digital (72, 123, 80)-net over F₃, using

5 times m-reduction [i] based on digital (72, 128, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 64, 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]

(123−51, 123, 112)-Net over F₃ — Digital

Digital (72, 123, 112)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−51, 123, 1059)-Net in Base 3 — Upper bound on s

There is no (72, 123, 1060)-net in base 3, because

1 times m-reduction [i] would yield (72, 122, 1060)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16351 367611 925563 358176 324477 391153 327337 075697 759586 131017 > 3¹²² [i]