MinT - Best Known (135−87, 135, s)-Nets in Base 3

Best Known (135−87, 135, s)-Nets in Base 3

(135−87, 135, 48)-Net over F₃ — Constructive and digital

Digital (48, 135, 48)-net over F₃, using

t-expansion [i] based on digital (45, 135, 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]

(135−87, 135, 56)-Net over F₃ — Digital

Digital (48, 135, 56)-net over F₃, using

t-expansion [i] based on digital (40, 135, 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]

(135−87, 135, 163)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 135, 164)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁵, 164, F₃, 87) (dual of [164, 29, 88]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹³⁴, 150, S₃, 87), but
    - the linear programming bound shows that MÂ â‰¥ 1998 191076 053927 805178 143382 564872 985089 052204 487349 698881 740338 723055 870219 / 227413 116931 > 3¹³⁴ [i]
  2. OA(3²⁹, 164, S₃, 14), but
    - discarding factors would yield OA(3²⁹, 163, S₃, 14), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 69 685080 318363 > 3²⁹ [i]

(135−87, 135, 219)-Net in Base 3 — Upper bound on s

There is no (48, 135, 220)-net in base 3, because

1 times m-reduction [i] would yield (48, 134, 220)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9887 475553 914173 061073 902333 154526 733132 982790 366654 599646 003025 > 3¹³⁴ [i]