MinT - Best Known (47, 129, s)-Nets in Base 3

Best Known (47, 129, s)-Nets in Base 3

(47, 129, 48)-Net over F₃ — Constructive and digital

Digital (47, 129, 48)-net over F₃, using

t-expansion [i] based on digital (45, 129, 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]

(47, 129, 56)-Net over F₃ — Digital

Digital (47, 129, 56)-net over F₃, using

t-expansion [i] based on digital (40, 129, 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]

(47, 129, 170)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 129, 171)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁹, 171, F₃, 82) (dual of [171, 42, 83]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹²⁸, 149, S₃, 82), but
    - the linear programming bound shows that MÂ â‰¥ 33 761969 841965 313899 084622 936677 465357 394104 583425 843880 075496 568722 581771 / 1 948280 920300 > 3¹²⁸ [i]
  2. OA(3⁴², 171, S₃, 22), but
    - discarding factors would yield OA(3⁴², 168, S₃, 22), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 114 464714 711551 910433 > 3⁴² [i]

(47, 129, 217)-Net in Base 3 — Upper bound on s

There is no (47, 129, 218)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 155412 305309 330943 769021 491808 201666 589123 936684 825105 902245 > 3¹²⁹ [i]