MinT - Best Known (50, 50+85, s)-Nets in Base 3

Best Known (50, 50+85, s)-Nets in Base 3

(50, 50+85, 48)-Net over F₃ — Constructive and digital

Digital (50, 135, 48)-net over F₃, using

t-expansion [i] based on digital (45, 135, 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, 50+85, 64)-Net over F₃ — Digital

Digital (50, 135, 64)-net over F₃, using

t-expansion [i] based on digital (49, 135, 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, 50+85, 230)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 135, 231)-net over F₃, because

1 times m-reduction [i] would yield digital (50, 134, 231)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³⁴, 231, F₃, 84) (dual of [231, 97, 85]-code), but
  - residual code [i] would yield OA(3⁵⁰, 146, S₃, 28), but
    - the linear programming bound shows that MÂ â‰¥ 1 005327 361531 234711 866975 289182 466989 734041 755525 / 1 282251 632725 448090 129998 > 3⁵⁰ [i]

(50, 50+85, 235)-Net in Base 3 — Upper bound on s

There is no (50, 135, 236)-net in base 3, because

1 times m-reduction [i] would yield (50, 134, 236)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8977 264061 137218 663857 761256 390984 091748 905809 734684 762515 641769 > 3¹³⁴ [i]