MinT - Best Known (56, 56+98, s)-Nets in Base 3

Best Known (56, 56+98, s)-Nets in Base 3

(56, 56+98, 48)-Net over F₃ — Constructive and digital

Digital (56, 154, 48)-net over F₃, using

t-expansion [i] based on digital (45, 154, 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]

(56, 56+98, 64)-Net over F₃ — Digital

Digital (56, 154, 64)-net over F₃, using

t-expansion [i] based on digital (49, 154, 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]

(56, 56+98, 246)-Net over F₃ — Upper bound on s (digital)

There is no digital (56, 154, 247)-net over F₃, because

2 times m-reduction [i] would yield digital (56, 152, 247)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵², 247, F₃, 96) (dual of [247, 95, 97]-code), but
  - residual code [i] would yield OA(3⁵⁶, 150, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 1 042121 495058 696990 589459 594226 638528 877701 897049 834059 577395 312500 / 1911 837015 144775 294785 556339 011193 770361 > 3⁵⁶ [i]

(56, 56+98, 256)-Net in Base 3 — Upper bound on s

There is no (56, 154, 257)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 33 342250 088168 859896 608468 032639 171499 644844 634708 724173 748074 748943 450947 > 3¹⁵⁴ [i]