MinT - Best Known (231−99, 231, s)-Nets in Base 3

Best Known (231−99, 231, s)-Nets in Base 3

(231−99, 231, 128)-Net over F₃ — Constructive and digital

Digital (132, 231, 128)-net over F₃, using

7 times m-reduction [i] based on digital (132, 238, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 119, 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]

(231−99, 231, 175)-Net over F₃ — Digital

Digital (132, 231, 175)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−99, 231, 1611)-Net in Base 3 — Upper bound on s

There is no (132, 231, 1612)-net in base 3, because

1 times m-reduction [i] would yield (132, 230, 1612)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 56 313428 087319 415717 165787 035373 796125 383029 881152 988212 282632 721808 031863 045563 837240 896144 396072 940180 738713 > 3²³⁰ [i]