MinT - Best Known (136−87, 136, s)-Nets in Base 3

Best Known (136−87, 136, s)-Nets in Base 3

(136−87, 136, 48)-Net over F₃ — Constructive and digital

Digital (49, 136, 48)-net over F₃, using

t-expansion [i] based on digital (45, 136, 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]

(136−87, 136, 64)-Net over F₃ — Digital

Digital (49, 136, 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]

(136−87, 136, 197)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 136, 198)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁶, 198, F₃, 87) (dual of [198, 62, 88]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹³⁵, 164, F₃, 87) (dual of [164, 29, 88]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(3¹³⁴, 150, S₃, 87), but
        the linear programming bound shows that MÂ â‰¥ 1998 191076 053927 805178 143382 564872 985089 052204 487349 698881 740338 723055 870219 / 227413 116931 > 3¹³⁴ [i]
      2. OA(3²⁹, 164, S₃, 14), but
        discarding factors would yield OA(3²⁹, 163, S₃, 14), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 69 685080 318363 > 3²⁹ [i]
  2. OA(3⁶², 198, S₃, 34), but
    - discarding factors would yield OA(3⁶², 187, S₃, 34), but
      - the linear programming bound shows that MÂ â‰¥ 86 582238 519540 759403 959995 312746 663126 603948 166833 519820 296875 / 223 323463 041708 430754 736757 380607 > 3⁶² [i]

(136−87, 136, 225)-Net in Base 3 — Upper bound on s

There is no (49, 136, 226)-net in base 3, because

1 times m-reduction [i] would yield (49, 135, 226)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26628 878185 200793 541137 584017 668580 859816 459511 287403 460269 681089 > 3¹³⁵ [i]