MinT - Best Known (136, 136+45, s)-Nets in Base 3

Best Known (136, 136+45, s)-Nets in Base 3

(136, 136+45, 328)-Net over F₃ — Constructive and digital

Digital (136, 181, 328)-net over F₃, using

3¹ times duplication [i] based on digital (135, 180, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 45, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(136, 136+45, 802)-Net over F₃ — Digital

Digital (136, 181, 802)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+45, 36248)-Net in Base 3 — Upper bound on s

There is no (136, 181, 36249)-net in base 3, because

1 times m-reduction [i] would yield (136, 180, 36249)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 76 220157 755140 557724 224706 908817 916476 440402 832077 271103 082525 026248 134473 419585 448457 > 3¹⁸⁰ [i]