MinT - Best Known (196−133, 196, s)-Nets in Base 3

Best Known (196−133, 196, s)-Nets in Base 3

(196−133, 196, 48)-Net over F₃ — Constructive and digital

Digital (63, 196, 48)-net over F₃, using

t-expansion [i] based on digital (45, 196, 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]

(196−133, 196, 64)-Net over F₃ — Digital

Digital (63, 196, 64)-net over F₃, using

t-expansion [i] based on digital (49, 196, 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]

(196−133, 196, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 196, 199)-net over F₃, because

4 times m-reduction [i] would yield digital (63, 192, 199)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹², 199, F₃, 129) (dual of [199, 7, 130]-code), but
  - residual code [i] would yield linear OA(3⁶³, 69, F₃, 43) (dual of [69, 6, 44]-code), but
    - â€œMaâ€ bound on codes from Brouwerâ€™s database [i]

(196−133, 196, 265)-Net in Base 3 — Upper bound on s

There is no (63, 196, 266)-net in base 3, because

1 times m-reduction [i] would yield (63, 195, 266)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1130 861887 532176 703983 498358 634047 191184 131182 208638 354581 855393 703446 500566 857816 606618 915205 > 3¹⁹⁵ [i]