MinT - Best Known (68, 68+55, s)-Nets in Base 3

Best Known (68, 68+55, s)-Nets in Base 3

(68, 68+55, 68)-Net over F₃ — Constructive and digital

Digital (68, 123, 68)-net over F₃, using

1 times m-reduction [i] based on digital (68, 124, 68)-net over F₃, using
- trace code for nets [i] based on digital (6, 62, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]

(68, 68+55, 91)-Net over F₃ — Digital

Digital (68, 123, 91)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+55, 755)-Net in Base 3 — Upper bound on s

There is no (68, 123, 756)-net in base 3, because

1 times m-reduction [i] would yield (68, 122, 756)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16195 051632 540465 808254 361039 117617 336866 850517 375642 477617 > 3¹²² [i]