MinT - Best Known (123−57, 123, s)-Nets in Base 3

Best Known (123−57, 123, s)-Nets in Base 3

(123−57, 123, 60)-Net over F₃ — Constructive and digital

Digital (66, 123, 60)-net over F₃, using

1 times m-reduction [i] based on digital (66, 124, 60)-net over F₃, using
- trace code for nets [i] based on digital (4, 62, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]

(123−57, 123, 82)-Net over F₃ — Digital

Digital (66, 123, 82)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−57, 123, 650)-Net in Base 3 — Upper bound on s

There is no (66, 123, 651)-net in base 3, because

1 times m-reduction [i] would yield (66, 122, 651)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16592 837498 851435 358693 211277 603055 185096 451558 262556 173225 > 3¹²² [i]