MinT - Best Known (186−125, 186, s)-Nets in Base 3

Best Known (186−125, 186, s)-Nets in Base 3

(186−125, 186, 48)-Net over F₃ — Constructive and digital

Digital (61, 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]

(186−125, 186, 64)-Net over F₃ — Digital

Digital (61, 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]

(186−125, 186, 195)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 186, 196)-net over F₃, because

2 times m-reduction [i] would yield digital (61, 184, 196)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁴, 196, F₃, 123) (dual of [196, 12, 124]-code), but
  - residual code [i] would yield OA(3⁶¹, 72, S₃, 41), but
    - the linear programming bound shows that MÂ â‰¥ 41997 386805 997720 199850 152210 995911 / 228095 > 3⁶¹ [i]

(186−125, 186, 260)-Net in Base 3 — Upper bound on s

There is no (61, 186, 261)-net in base 3, because

1 times m-reduction [i] would yield (61, 185, 261)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 20760 607854 796677 499508 680735 396404 898451 561034 913242 974404 908711 377255 071359 693766 873377 > 3¹⁸⁵ [i]