MinT - Best Known (49, 144, s)-Nets in Base 3

Best Known (49, 144, s)-Nets in Base 3

(49, 144, 48)-Net over F₃ — Constructive and digital

Digital (49, 144, 48)-net over F₃, using

t-expansion [i] based on digital (45, 144, 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, 144, 64)-Net over F₃ — Digital

Digital (49, 144, 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, 144, 167)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 144, 168)-net over F₃, because

1 times m-reduction [i] would yield digital (49, 143, 168)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴³, 168, F₃, 94) (dual of [168, 25, 95]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁴², 156, F₃, 94) (dual of [156, 14, 95]-code), but
      - constructionÂ Y1 [i] would yield
        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]
        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]
    2. OA(3²⁵, 168, S₃, 12), but
      - discarding factors would yield OA(3²⁵, 148, S₃, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 861032 991633 > 3²⁵ [i]

(49, 144, 216)-Net in Base 3 — Upper bound on s

There is no (49, 144, 217)-net in base 3, because

1 times m-reduction [i] would yield (49, 143, 217)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 184 530253 174878 553314 516712 282799 197694 257088 880545 781236 795298 008907 > 3¹⁴³ [i]