MinT - Best Known (134−85, 134, s)-Nets in Base 3

Best Known (134−85, 134, s)-Nets in Base 3

(134−85, 134, 48)-Net over F₃ — Constructive and digital

Digital (49, 134, 48)-net over F₃, using

t-expansion [i] based on digital (45, 134, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(134−85, 134, 64)-Net over F₃ — Digital

Digital (49, 134, 64)-net over F₃, using

net from sequence [i] based on digital (49, 63)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
  - a curve from the manYPoints database [i]

(134−85, 134, 218)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 134, 219)-net over F₃, because

1 times m-reduction [i] would yield digital (49, 133, 219)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³³, 219, F₃, 84) (dual of [219, 86, 85]-code), but
  - residual code [i] would yield OA(3⁴⁹, 134, S₃, 28), but
    - the linear programming bound shows that MÂ â‰¥ 430073 037911 139946 191996 083081 250824 628251 158946 484375 / 1 673823 880993 304164 747289 137687 > 3⁴⁹ [i]

(134−85, 134, 228)-Net in Base 3 — Upper bound on s

There is no (49, 134, 229)-net in base 3, because

1 times m-reduction [i] would yield (49, 133, 229)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3012 787058 047654 628325 945759 683973 576846 780980 385591 034643 075889 > 3¹³³ [i]