MinT - Best Known (49, 146, s)-Nets in Base 3

Best Known (49, 146, s)-Nets in Base 3

(49, 146, 48)-Net over F₃ — Constructive and digital

Digital (49, 146, 48)-net over F₃, using

t-expansion [i] based on digital (45, 146, 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, 146, 64)-Net over F₃ — Digital

Digital (49, 146, 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, 146, 165)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 146, 166)-net over F₃, because

1 times m-reduction [i] would yield digital (49, 145, 166)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁵, 166, F₃, 96) (dual of [166, 21, 97]-code), but
  - residual code [i] would yield OA(3⁴⁹, 69, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 216581 757923 721967 421601 778838 750385 / 815783 089121 > 3⁴⁹ [i]

(49, 146, 214)-Net in Base 3 — Upper bound on s

There is no (49, 146, 215)-net in base 3, because

1 times m-reduction [i] would yield (49, 145, 215)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1620 123905 915841 556424 579036 253414 495865 104073 665674 842624 100810 972321 > 3¹⁴⁵ [i]