MinT - Best Known (49, 49+103, s)-Nets in Base 3

Best Known (49, 49+103, s)-Nets in Base 3

(49, 49+103, 48)-Net over F₃ — Constructive and digital

Digital (49, 152, 48)-net over F₃, using

t-expansion [i] based on digital (45, 152, 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]

(49, 49+103, 64)-Net over F₃ — Digital

Digital (49, 152, 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]

(49, 49+103, 157)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 152, 158)-net over F₃, because

4 times m-reduction [i] would yield digital (49, 148, 158)-net over F₃, but
- extracting embedded orthogonal array [i] would yield 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]

(49, 49+103, 209)-Net in Base 3 — Upper bound on s

There is no (49, 152, 210)-net in base 3, because

1 times m-reduction [i] would yield (49, 151, 210)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 114590 524275 360406 140982 173640 611227 309934 843002 838560 396455 352750 660657 > 3¹⁵¹ [i]