MinT - Best Known (48, 48+85, s)-Nets in Base 3

Best Known (48, 48+85, s)-Nets in Base 3

(48, 48+85, 48)-Net over F₃ — Constructive and digital

Digital (48, 133, 48)-net over F₃, using

t-expansion [i] based on digital (45, 133, 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]

(48, 48+85, 56)-Net over F₃ — Digital

Digital (48, 133, 56)-net over F₃, using

t-expansion [i] based on digital (40, 133, 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]

(48, 48+85, 200)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 133, 201)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³³, 201, F₃, 85) (dual of [201, 68, 86]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹³², 163, F₃, 85) (dual of [163, 31, 86]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(3¹³¹, 147, S₃, 85), but
        the linear programming bound shows that MÂ â‰¥ 2 959964 272368 355385 131019 565631 322230 864774 663367 805086 508948 978796 443201 / 9282 754000 > 3¹³¹ [i]
      2. OA(3³¹, 163, S₃, 16), but
        discarding factors would yield OA(3³¹, 136, S₃, 16), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 621 653656 785825 > 3³¹ [i]
  2. OA(3⁶⁸, 201, S₃, 38), but
    - discarding factors would yield 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]

(48, 48+85, 221)-Net in Base 3 — Upper bound on s

There is no (48, 133, 222)-net in base 3, because

1 times m-reduction [i] would yield (48, 132, 222)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 981 584631 427719 332754 876330 731889 625965 254788 665125 744333 217549 > 3¹³² [i]