MinT - Best Known (38−12, 38, s)-Nets in Base 3

Best Known (38−12, 38, s)-Nets in Base 3

(38−12, 38, 84)-Net over F₃ — Constructive and digital

Digital (26, 38, 84)-net over F₃, using

1 times m-reduction [i] based on digital (26, 39, 84)-net over F₃, using
- trace code for nets [i] based on digital (0, 13, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]

(38−12, 38, 124)-Net over F₃ — Digital

Digital (26, 38, 124)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³⁸, 124, F₃, 12) (dual of [124, 86, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3³⁸, 134, F₃, 12) (dual of [134, 96, 13]-code), using
  - a code from Grasslâ€™s database [i]

(38−12, 38, 1568)-Net in Base 3 — Upper bound on s

There is no (26, 38, 1569)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 354222 824772 007961 > 3³⁸ [i]