MinT - Best Known (51, 150, s)-Nets in Base 3

Best Known (51, 150, s)-Nets in Base 3

(51, 150, 48)-Net over F₃ — Constructive and digital

Digital (51, 150, 48)-net over F₃, using

t-expansion [i] based on digital (45, 150, 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]

(51, 150, 64)-Net over F₃ — Digital

Digital (51, 150, 64)-net over F₃, using

t-expansion [i] based on digital (49, 150, 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]

(51, 150, 173)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 150, 174)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁰, 174, F₃, 99) (dual of [174, 24, 100]-code), but
- residual code [i] would yield OA(3⁵¹, 74, S₃, 33), but
  - the linear programming bound shows that MÂ â‰¥ 123 263670 768810 287673 347725 757839 680807 / 56 048467 475000 > 3⁵¹ [i]

(51, 150, 224)-Net in Base 3 — Upper bound on s

There is no (51, 150, 225)-net in base 3, because

1 times m-reduction [i] would yield (51, 149, 225)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 131229 497666 905798 066676 318663 772816 002106 861938 791945 090060 161113 546499 > 3¹⁴⁹ [i]