MinT - Best Known (153−97, 153, s)-Nets in Base 3

Best Known (153−97, 153, s)-Nets in Base 3

(153−97, 153, 48)-Net over F₃ — Constructive and digital

Digital (56, 153, 48)-net over F₃, using

t-expansion [i] based on digital (45, 153, 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]

(153−97, 153, 64)-Net over F₃ — Digital

Digital (56, 153, 64)-net over F₃, using

t-expansion [i] based on digital (49, 153, 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]

(153−97, 153, 246)-Net over F₃ — Upper bound on s (digital)

There is no digital (56, 153, 247)-net over F₃, because

1 times m-reduction [i] would yield digital (56, 152, 247)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵², 247, F₃, 96) (dual of [247, 95, 97]-code), but
  - residual code [i] would yield OA(3⁵⁶, 150, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 1 042121 495058 696990 589459 594226 638528 877701 897049 834059 577395 312500 / 1911 837015 144775 294785 556339 011193 770361 > 3⁵⁶ [i]

(153−97, 153, 259)-Net in Base 3 — Upper bound on s

There is no (56, 153, 260)-net in base 3, because

1 times m-reduction [i] would yield (56, 152, 260)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 784964 917217 903061 018309 769252 166365 589912 806973 384537 219202 683140 008321 > 3¹⁵² [i]