MinT - Best Known (189−127, 189, s)-Nets in Base 3

Best Known (189−127, 189, s)-Nets in Base 3

(189−127, 189, 48)-Net over F₃ — Constructive and digital

Digital (62, 189, 48)-net over F₃, using

t-expansion [i] based on digital (45, 189, 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]

(189−127, 189, 64)-Net over F₃ — Digital

Digital (62, 189, 64)-net over F₃, using

t-expansion [i] based on digital (49, 189, 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]

(189−127, 189, 195)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 189, 196)-net over F₃, because

1 times m-reduction [i] would yield digital (62, 188, 196)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁸, 196, F₃, 126) (dual of [196, 8, 127]-code), but
  - residual code [i] would yield linear OA(3⁶², 69, F₃, 42) (dual of [69, 7, 43]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(189−127, 189, 264)-Net in Base 3 — Upper bound on s

There is no (62, 189, 265)-net in base 3, because

1 times m-reduction [i] would yield (62, 188, 265)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 551792 921709 981939 338166 245282 146263 439966 409287 904360 555599 782529 517726 548499 847822 519243 > 3¹⁸⁸ [i]