MinT - Best Known (157−107, 157, s)-Nets in Base 3

Best Known (157−107, 157, s)-Nets in Base 3

(157−107, 157, 48)-Net over F₃ — Constructive and digital

Digital (50, 157, 48)-net over F₃, using

t-expansion [i] based on digital (45, 157, 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]

(157−107, 157, 64)-Net over F₃ — Digital

Digital (50, 157, 64)-net over F₃, using

t-expansion [i] based on digital (49, 157, 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]

(157−107, 157, 161)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 157, 162)-net over F₃, because

5 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]

(157−107, 157, 212)-Net in Base 3 — Upper bound on s

There is no (50, 157, 213)-net in base 3, because

1 times m-reduction [i] would yield (50, 156, 213)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 283 433163 688457 732471 942452 133853 618968 582484 874013 085527 500983 520110 182907 > 3¹⁵⁶ [i]