MinT - Best Known (234−51, 234, s)-Nets in Base 3

Best Known (234−51, 234, s)-Nets in Base 3

(234−51, 234, 688)-Net over F₃ — Constructive and digital

Digital (183, 234, 688)-net over F₃, using

3² times duplication [i] based on digital (181, 232, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(234−51, 234, 1690)-Net over F₃ — Digital

Digital (183, 234, 1690)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−51, 234, 142328)-Net in Base 3 — Upper bound on s

There is no (183, 234, 142329)-net in base 3, because

1 times m-reduction [i] would yield (183, 233, 142329)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1476 752841 371275 779001 139390 824208 001612 427183 802062 134729 925292 720157 777743 319816 497951 700069 494300 864147 110419 > 3²³³ [i]