MinT - Best Known (47, 47+81, s)-Nets in Base 3

Best Known (47, 47+81, s)-Nets in Base 3

(47, 47+81, 48)-Net over F₃ — Constructive and digital

Digital (47, 128, 48)-net over F₃, using

t-expansion [i] based on digital (45, 128, 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]

(47, 47+81, 56)-Net over F₃ — Digital

Digital (47, 128, 56)-net over F₃, using

t-expansion [i] based on digital (40, 128, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(47, 47+81, 175)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 128, 176)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁸, 176, F₃, 81) (dual of [176, 48, 82]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹²⁷, 150, S₃, 81), but
    - the linear programming bound shows that MÂ â‰¥ 34 591649 218027 645930 654651 997419 100907 498551 201171 263936 373038 687154 245563 / 8 746966 630402 > 3¹²⁷ [i]
  2. OA(3⁴⁸, 176, S₃, 26), but
    - discarding factors would yield OA(3⁴⁸, 170, S₃, 26), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 84876 685847 451322 200873 > 3⁴⁸ [i]

(47, 47+81, 220)-Net in Base 3 — Upper bound on s

There is no (47, 128, 221)-net in base 3, because

1 times m-reduction [i] would yield (47, 127, 221)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 034416 545562 885627 535947 609371 258141 019941 722173 177322 525409 > 3¹²⁷ [i]