MinT - Best Known (90, 250, s)-Nets in Base 3

Best Known (90, 250, s)-Nets in Base 3

Digital (90, 250, 64)-net over F₃, using

t-expansion [i] based on digital (89, 250, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

Digital (90, 250, 96)-net over F₃, using

There is no digital (90, 250, 394)-net over F₃, because

1 times m-reduction [i] would yield digital (90, 249, 394)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁹, 394, F₃, 159) (dual of [394, 145, 160]-code), but
  - residual code [i] would yield linear OA(3⁹⁰, 234, F₃, 53) (dual of [234, 144, 54]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁹, 233, F₃, 52) (dual of [233, 144, 53]-code), but
      - the Johnson bound shows that NÂ â‰¤ 495 041572 388827 382337 917134 411719 483784 605757 670798 500294 630099 278193 < 3¹⁴⁴ [i]

There is no (90, 250, 400)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 209023 282752 628743 712445 404612 906887 460458 348347 958213 063400 436717 502708 917981 220576 808813 173780 700457 695321 592364 164097 > 3²⁵⁰ [i]