MinT - Best Known (50, 50+98, s)-Nets in Base 3

Best Known (50, 50+98, s)-Nets in Base 3

(50, 50+98, 48)-Net over F₃ — Constructive and digital

Digital (50, 148, 48)-net over F₃, using

t-expansion [i] based on digital (45, 148, 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+98, 64)-Net over F₃ — Digital

Digital (50, 148, 64)-net over F₃, using

t-expansion [i] based on digital (49, 148, 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+98, 176)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 148, 177)-net over F₃, because

2 times m-reduction [i] would yield digital (50, 146, 177)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁶, 177, F₃, 96) (dual of [177, 31, 97]-code), but
  - residual code [i] would yield OA(3⁵⁰, 80, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 17966 137866 413855 600211 208294 867018 069814 838765 / 21749 437213 575033 309776 > 3⁵⁰ [i]

(50, 50+98, 218)-Net in Base 3 — Upper bound on s

There is no (50, 148, 219)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 43105 064385 331788 801617 790156 097721 863262 376046 664046 689130 636266 515671 > 3¹⁴⁸ [i]