MinT - Best Known (47, 47+80, s)-Nets in Base 3

Best Known (47, 47+80, s)-Nets in Base 3

(47, 47+80, 48)-Net over F₃ — Constructive and digital

Digital (47, 127, 48)-net over F₃, using

t-expansion [i] based on digital (45, 127, 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, 47+80, 56)-Net over F₃ — Digital

Digital (47, 127, 56)-net over F₃, using

t-expansion [i] based on digital (40, 127, 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, 47+80, 177)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 127, 178)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁷, 178, F₃, 80) (dual of [178, 51, 81]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹²⁶, 150, S₃, 80), but
    - the linear programming bound shows that MÂ â‰¥ 1 744284 668385 163500 346306 269658 104769 929699 691581 439318 970712 105752 702784 / 1 186440 433087 > 3¹²⁶ [i]
  2. OA(3⁵¹, 178, S₃, 28), but
    - discarding factors would yield OA(3⁵¹, 172, S₃, 28), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2 263924 665338 043697 613937 > 3⁵¹ [i]

(47, 47+80, 220)-Net in Base 3 — Upper bound on s

There is no (47, 127, 221)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 034416 545562 885627 535947 609371 258141 019941 722173 177322 525409 > 3¹²⁷ [i]