MinT - Best Known (48, 48+98, s)-Nets in Base 3

Best Known (48, 48+98, s)-Nets in Base 3

(48, 48+98, 48)-Net over F₃ — Constructive and digital

Digital (48, 146, 48)-net over F₃, using

t-expansion [i] based on digital (45, 146, 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, 48+98, 56)-Net over F₃ — Digital

Digital (48, 146, 56)-net over F₃, using

t-expansion [i] based on digital (40, 146, 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, 48+98, 155)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 146, 156)-net over F₃, because

4 times m-reduction [i] would yield digital (48, 142, 156)-net over F₃, but
- 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, 48+98, 207)-Net in Base 3 — Upper bound on s

There is no (48, 146, 208)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5222 119428 600862 316307 616100 016168 001208 995726 942145 599359 620592 038817 > 3¹⁴⁶ [i]