MinT - Best Known (51, 51+89, s)-Nets in Base 3

Best Known (51, 51+89, s)-Nets in Base 3

(51, 51+89, 48)-Net over F₃ — Constructive and digital

Digital (51, 140, 48)-net over F₃, using

t-expansion [i] based on digital (45, 140, 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, 51+89, 64)-Net over F₃ — Digital

Digital (51, 140, 64)-net over F₃, using

t-expansion [i] based on digital (49, 140, 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, 51+89, 233)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 140, 234)-net over F₃, because

2 times m-reduction [i] would yield digital (51, 138, 234)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³⁸, 234, F₃, 87) (dual of [234, 96, 88]-code), but
  - residual code [i] would yield OA(3⁵¹, 146, S₃, 29), but
    - the linear programming bound shows that MÂ â‰¥ 5126 774601 249175 366045 349858 741246 206423 340601 352320 / 2311 172455 881906 807675 878437 > 3⁵¹ [i]

(51, 51+89, 236)-Net in Base 3 — Upper bound on s

There is no (51, 140, 237)-net in base 3, because

1 times m-reduction [i] would yield (51, 139, 237)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 241081 966377 953175 766110 047600 960055 715308 965329 250981 111413 683697 > 3¹³⁹ [i]