MinT - Best Known (62, 186, s)-Nets in Base 3

Best Known (62, 186, s)-Nets in Base 3

(62, 186, 48)-Net over F₃ — Constructive and digital

Digital (62, 186, 48)-net over F₃, using

t-expansion [i] based on digital (45, 186, 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]

(62, 186, 64)-Net over F₃ — Digital

Digital (62, 186, 64)-net over F₃, using

t-expansion [i] based on digital (49, 186, 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]

(62, 186, 199)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 186, 200)-net over F₃, because

1 times m-reduction [i] would yield digital (62, 185, 200)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁵, 200, F₃, 123) (dual of [200, 15, 124]-code), but
  - residual code [i] would yield OA(3⁶², 76, S₃, 41), but
    - the linear programming bound shows that MÂ â‰¥ 347660 486804 616889 071607 220289 701250 / 818741 > 3⁶² [i]

(62, 186, 266)-Net in Base 3 — Upper bound on s

There is no (62, 186, 267)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 67130 026070 885443 952951 463407 853497 466841 465511 142365 945775 839475 127095 648259 435418 385213 > 3¹⁸⁶ [i]