MinT - Best Known (49, 49+88, s)-Nets in Base 3

Best Known (49, 49+88, s)-Nets in Base 3

(49, 49+88, 48)-Net over F₃ — Constructive and digital

Digital (49, 137, 48)-net over F₃, using

t-expansion [i] based on digital (45, 137, 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]

(49, 49+88, 64)-Net over F₃ — Digital

Digital (49, 137, 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]

(49, 49+88, 197)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 137, 198)-net over F₃, because

1 times m-reduction [i] would yield digital (49, 136, 198)-net over F₃, but
- 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
        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]
        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]

(49, 49+88, 223)-Net in Base 3 — Upper bound on s

There is no (49, 137, 224)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 266846 708634 694644 431460 539133 465720 208703 483418 391363 759037 977857 > 3¹³⁷ [i]