MinT - Best Known (50, 149, s)-Nets in Base 3

Best Known (50, 149, s)-Nets in Base 3

(50, 149, 48)-Net over F₃ — Constructive and digital

Digital (50, 149, 48)-net over F₃, using

t-expansion [i] based on digital (45, 149, 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, 149, 64)-Net over F₃ — Digital

Digital (50, 149, 64)-net over F₃, using

t-expansion [i] based on digital (49, 149, 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, 149, 165)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 149, 166)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁹, 166, F₃, 99) (dual of [166, 17, 100]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹⁴⁸, 158, F₃, 99) (dual of [158, 10, 100]-code), but
    - residual code [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]
  2. OA(3¹⁷, 166, S₃, 8), but
    - discarding factors would yield OA(3¹⁷, 119, S₃, 8), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 129 270891 > 3¹⁷ [i]

(50, 149, 218)-Net in Base 3 — Upper bound on s

There is no (50, 149, 219)-net in base 3, because

1 times m-reduction [i] would yield (50, 148, 219)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 43105 064385 331788 801617 790156 097721 863262 376046 664046 689130 636266 515671 > 3¹⁴⁸ [i]