MinT - Best Known (45, 45+74, s)-Nets in Base 3

Best Known (45, 45+74, s)-Nets in Base 3

(45, 45+74, 48)-Net over F₃ — Constructive and digital

Digital (45, 119, 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]

(45, 45+74, 56)-Net over F₃ — Digital

Digital (45, 119, 56)-net over F₃, using

t-expansion [i] based on digital (40, 119, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(45, 45+74, 186)-Net over F₃ — Upper bound on s (digital)

There is no digital (45, 119, 187)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹¹⁹, 187, F₃, 74) (dual of [187, 68, 75]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹¹⁸, 149, S₃, 74), but
    - the linear programming bound shows that MÂ â‰¥ 14631 013686 142997 953866 980562 476233 047154 143146 692997 967075 486096 911649 489501 / 62 737137 135696 970925 > 3¹¹⁸ [i]
  2. OA(3⁶⁸, 187, S₃, 38), but
    - the linear programming bound shows that MÂ â‰¥ 4 185011 256460 307952 313976 653889 690485 861995 490330 218229 391251 955691 449885 824042 401792 / 14773 934323 743575 783185 881134 593137 934301 575737 911449 > 3⁶⁸ [i]

(45, 45+74, 216)-Net in Base 3 — Upper bound on s

There is no (45, 119, 217)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 654 638524 372023 414877 758292 895346 269335 981374 255100 133283 > 3¹¹⁹ [i]