MinT - Best Known (152−101, 152, s)-Nets in Base 3

Best Known (152−101, 152, s)-Nets in Base 3

(152−101, 152, 48)-Net over F₃ — Constructive and digital

Digital (51, 152, 48)-net over F₃, using

t-expansion [i] based on digital (45, 152, 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]

(152−101, 152, 64)-Net over F₃ — Digital

Digital (51, 152, 64)-net over F₃, using

t-expansion [i] based on digital (49, 152, 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]

(152−101, 152, 173)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 152, 174)-net over F₃, because

2 times m-reduction [i] would yield digital (51, 150, 174)-net over F₃, but
- 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]

(152−101, 152, 222)-Net in Base 3 — Upper bound on s

There is no (51, 152, 223)-net in base 3, because

1 times m-reduction [i] would yield (51, 151, 223)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 146912 801671 120993 229376 302165 347533 482272 848210 826840 356231 677099 512157 > 3¹⁵¹ [i]