MinT - Best Known (52, 52+103, s)-Nets in Base 3

Best Known (52, 52+103, s)-Nets in Base 3

(52, 52+103, 48)-Net over F₃ — Constructive and digital

Digital (52, 155, 48)-net over F₃, using

t-expansion [i] based on digital (45, 155, 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]

(52, 52+103, 64)-Net over F₃ — Digital

Digital (52, 155, 64)-net over F₃, using

t-expansion [i] based on digital (49, 155, 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]

(52, 52+103, 174)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 155, 175)-net over F₃, because

1 times m-reduction [i] would yield digital (52, 154, 175)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁴, 175, F₃, 102) (dual of [175, 21, 103]-code), but
  - residual code [i] would yield OA(3⁵², 72, S₃, 34), but
    - the linear programming bound shows that MÂ â‰¥ 23373 085764 731132 483065 978750 735767 / 2987 457005 > 3⁵² [i]

(52, 52+103, 226)-Net in Base 3 — Upper bound on s

There is no (52, 155, 227)-net in base 3, because

1 times m-reduction [i] would yield (52, 154, 227)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30 517907 170621 009863 270979 237760 703065 982556 875572 327525 424439 586349 367755 > 3¹⁵⁴ [i]