MinT - Best Known (132−85, 132, s)-Nets in Base 3

Best Known (132−85, 132, s)-Nets in Base 3

(132−85, 132, 48)-Net over F₃ — Constructive and digital

Digital (47, 132, 48)-net over F₃, using

t-expansion [i] based on digital (45, 132, 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]

(132−85, 132, 56)-Net over F₃ — Digital

Digital (47, 132, 56)-net over F₃, using

t-expansion [i] based on digital (40, 132, 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]

(132−85, 132, 162)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 132, 163)-net over F₃, because

1 times m-reduction [i] would yield digital (47, 131, 163)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³¹, 163, F₃, 84) (dual of [163, 32, 85]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(3¹³⁰, 147, S₃, 84), but
      - the linear programming bound shows that MÂ â‰¥ 610951 874237 245179 799757 396293 504258 972145 948730 720728 290459 967541 402306 / 4917 657235 > 3¹³⁰ [i]
    2. OA(3³², 163, S₃, 16), but
      - discarding factors would yield OA(3³², 156, S₃, 16), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1906 607562 901809 > 3³² [i]

(132−85, 132, 215)-Net in Base 3 — Upper bound on s

There is no (47, 132, 216)-net in base 3, because

1 times m-reduction [i] would yield (47, 131, 216)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 366 162728 829372 076375 417821 613420 480978 715358 294129 629287 984145 > 3¹³¹ [i]