MinT - Best Known (50, 137, s)-Nets in Base 3

Best Known (50, 137, s)-Nets in Base 3

(50, 137, 48)-Net over F₃ — Constructive and digital

Digital (50, 137, 48)-net over F₃, using

t-expansion [i] based on digital (45, 137, 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]

(50, 137, 64)-Net over F₃ — Digital

Digital (50, 137, 64)-net over F₃, using

t-expansion [i] based on digital (49, 137, 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]

(50, 137, 222)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 137, 223)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁷, 223, F₃, 87) (dual of [223, 86, 88]-code), but
- residual code [i] would yield OA(3⁵⁰, 135, S₃, 29), but
  - 1 times truncation [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]

(50, 137, 232)-Net in Base 3 — Upper bound on s

There is no (50, 137, 233)-net in base 3, because

1 times m-reduction [i] would yield (50, 136, 233)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 82207 623651 688630 493239 212026 178377 394483 902501 316847 583228 256091 > 3¹³⁶ [i]