MinT - Best Known (50, 50+128, s)-Nets in Base 3

Best Known (50, 50+128, s)-Nets in Base 3

(50, 50+128, 48)-Net over F₃ — Constructive and digital

Digital (50, 178, 48)-net over F₃, using

t-expansion [i] based on digital (45, 178, 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]

(50, 50+128, 64)-Net over F₃ — Digital

Digital (50, 178, 64)-net over F₃, using

t-expansion [i] based on digital (49, 178, 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]

(50, 50+128, 161)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 178, 162)-net over F₃, because

26 times m-reduction [i] would yield digital (50, 152, 162)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵², 162, F₃, 102) (dual of [162, 10, 103]-code), but
  - residual code [i] would yield linear OA(3⁵⁰, 59, F₃, 34) (dual of [59, 9, 35]-code), but
    - 1 times truncation [i] would yield linear OA(3⁴⁹, 58, F₃, 33) (dual of [58, 9, 34]-code), but
      - residual code [i] would yield linear OA(3¹⁶, 24, F₃, 11) (dual of [24, 8, 12]-code), but
        â€œBSâ€ bound on codes from Brouwerâ€™s database [i]

(50, 50+128, 163)-Net in Base 3 — Upper bound on s

There is no (50, 178, 164)-net in base 3, because

19 times m-reduction [i] would yield (50, 159, 164)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁵⁹, 164, S₃, 109), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 589881 151426 658740 854227 725580 736348 849310 352832 644300 781946 246613 899173 590427 / 55 > 3¹⁵⁹ [i]