MinT - Best Known (49, 49+89, s)-Nets in Base 3

Best Known (49, 49+89, s)-Nets in Base 3

(49, 49+89, 48)-Net over F₃ — Constructive and digital

Digital (49, 138, 48)-net over F₃, using

t-expansion [i] based on digital (45, 138, 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+89, 64)-Net over F₃ — Digital

Digital (49, 138, 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+89, 186)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 138, 187)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁸, 187, F₃, 89) (dual of [187, 49, 90]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹³⁷, 161, F₃, 89) (dual of [161, 24, 90]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(3¹³⁶, 149, S₃, 89), but
        the linear programming bound shows that MÂ â‰¥ 6 385896 362423 839893 456186 755923 270794 744764 572661 578917 513381 685836 804369 / 75 686875 > 3¹³⁶ [i]
      2. OA(3²⁴, 161, S₃, 12), but
        discarding factors would yield OA(3²⁴, 124, S₃, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 293147 842993 > 3²⁴ [i]
  2. OA(3⁴⁹, 187, S₃, 26), but
    - discarding factors would yield OA(3⁴⁹, 184, S₃, 26), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 245469 510905 440390 144353 > 3⁴⁹ [i]

(49, 49+89, 223)-Net in Base 3 — Upper bound on s

There is no (49, 138, 224)-net in base 3, because

1 times m-reduction [i] would yield (49, 137, 224)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 266846 708634 694644 431460 539133 465720 208703 483418 391363 759037 977857 > 3¹³⁷ [i]