MinT - Best Known (188−128, 188, s)-Nets in Base 3

Best Known (188−128, 188, s)-Nets in Base 3

(188−128, 188, 48)-Net over F₃ — Constructive and digital

Digital (60, 188, 48)-net over F₃, using

t-expansion [i] based on digital (45, 188, 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]

(188−128, 188, 64)-Net over F₃ — Digital

Digital (60, 188, 64)-net over F₃, using

t-expansion [i] based on digital (49, 188, 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]

(188−128, 188, 190)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 188, 191)-net over F₃, because

5 times m-reduction [i] would yield digital (60, 183, 191)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸³, 191, F₃, 123) (dual of [191, 8, 124]-code), but
  - residual code [i] would yield OA(3⁶⁰, 67, S₃, 41), but
    - the linear programming bound shows that MÂ â‰¥ 367 404168 771298 835858 389852 553067 / 8575 > 3⁶⁰ [i]

(188−128, 188, 252)-Net in Base 3 — Upper bound on s

There is no (60, 188, 253)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 570391 668368 469606 812187 003972 484179 205201 176556 391514 614366 317201 744849 063478 349250 381569 > 3¹⁸⁸ [i]