MinT - Best Known (48, 142, s)-Nets in Base 3

Best Known (48, 142, s)-Nets in Base 3

(48, 142, 48)-Net over F₃ — Constructive and digital

Digital (48, 142, 48)-net over F₃, using

t-expansion [i] based on digital (45, 142, 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]

(48, 142, 56)-Net over F₃ — Digital

Digital (48, 142, 56)-net over F₃, using

t-expansion [i] based on digital (40, 142, 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]

(48, 142, 155)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 142, 156)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴², 156, F₃, 94) (dual of [156, 14, 95]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹⁴¹, 150, S₃, 94), but
    - the linear programming bound shows that MÂ â‰¥ 2 507699 731904 749483 156305 303283 721699 486095 458197 011266 969583 191489 627021 / 107749 > 3¹⁴¹ [i]
  2. OA(3¹⁴, 156, S₃, 6), but
    - discarding factors would yield OA(3¹⁴, 154, S₃, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 822665 > 3¹⁴ [i]

(48, 142, 210)-Net in Base 3 — Upper bound on s

There is no (48, 142, 211)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 60 896160 244306 309113 446140 625537 296442 515982 401868 498121 712100 004339 > 3¹⁴² [i]