MinT - Best Known (45, 45+75, s)-Nets in Base 3

Best Known (45, 45+75, s)-Nets in Base 3

(45, 45+75, 48)-Net over F₃ — Constructive and digital

Digital (45, 120, 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+75, 56)-Net over F₃ — Digital

Digital (45, 120, 56)-net over F₃, using

t-expansion [i] based on digital (40, 120, 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+75, 180)-Net over F₃ — Upper bound on s (digital)

There is no digital (45, 120, 181)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁰, 181, F₃, 75) (dual of [181, 61, 76]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹¹⁹, 147, S₃, 75), but
    - the linear programming bound shows that MÂ â‰¥ 16 137915 552723 745131 682214 391750 091116 522557 190679 736158 328855 711764 240731 / 23372 130007 496452 > 3¹¹⁹ [i]
  2. OA(3⁶¹, 181, S₃, 34), but
    - discarding factors would yield OA(3⁶¹, 176, S₃, 34), but
      - the linear programming bound shows that MÂ â‰¥ 812 356626 791711 438639 003017 318488 339021 708011 289038 629210 828125 / 5801 809430 490682 982977 100374 687357 > 3⁶¹ [i]

(45, 45+75, 216)-Net in Base 3 — Upper bound on s

There is no (45, 120, 217)-net in base 3, because

1 times m-reduction [i] would yield (45, 119, 217)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 654 638524 372023 414877 758292 895346 269335 981374 255100 133283 > 3¹¹⁹ [i]