MinT - Best Known (152−100, 152, s)-Nets in Base 3

Best Known (152−100, 152, s)-Nets in Base 3

(152−100, 152, 48)-Net over F₃ — Constructive and digital

Digital (52, 152, 48)-net over F₃, using

t-expansion [i] based on digital (45, 152, 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]

(152−100, 152, 64)-Net over F₃ — Digital

Digital (52, 152, 64)-net over F₃, using

t-expansion [i] based on digital (49, 152, 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]

(152−100, 152, 184)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 152, 185)-net over F₃, because

1 times m-reduction [i] would yield digital (52, 151, 185)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵¹, 185, F₃, 99) (dual of [185, 34, 100]-code), but
  - residual code [i] would yield OA(3⁵², 85, S₃, 33), but
    - the linear programming bound shows that MÂ â‰¥ 860342 754412 435094 906698 858579 674571 922984 984079 / 127652 777684 493779 532100 > 3⁵² [i]

(152−100, 152, 228)-Net in Base 3 — Upper bound on s

There is no (52, 152, 229)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 499383 927897 153073 464281 317727 815139 979099 926954 434049 931893 399945 635281 > 3¹⁵² [i]