MinT - Best Known (54, 54+94, s)-Nets in Base 3

Best Known (54, 54+94, s)-Nets in Base 3

(54, 54+94, 48)-Net over F₃ — Constructive and digital

Digital (54, 148, 48)-net over F₃, using

t-expansion [i] based on digital (45, 148, 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]

(54, 54+94, 64)-Net over F₃ — Digital

Digital (54, 148, 64)-net over F₃, using

t-expansion [i] based on digital (49, 148, 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]

(54, 54+94, 241)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 148, 242)-net over F₃, because

1 times m-reduction [i] would yield digital (54, 147, 242)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁷, 242, F₃, 93) (dual of [242, 95, 94]-code), but
  - residual code [i] would yield OA(3⁵⁴, 148, S₃, 31), but
    - the linear programming bound shows that MÂ â‰¥ 6 441601 373247 630548 677274 000331 028968 435027 093798 890625 / 110747 556584 570142 271976 733329 > 3⁵⁴ [i]

(54, 54+94, 248)-Net in Base 3 — Upper bound on s

There is no (54, 148, 249)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 45179 100490 003004 049095 857476 687529 372172 948634 042387 550737 036058 265547 > 3¹⁴⁸ [i]