MinT - Best Known (57, 204, s)-Nets in Base 4

Best Known (57, 204, s)-Nets in Base 4

Digital (57, 204, 66)-net over F₄, using

t-expansion [i] based on digital (49, 204, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (57, 204, 91)-net over F₄, using

t-expansion [i] based on digital (50, 204, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

There is no digital (57, 204, 340)-net over F₄, because

3 times m-reduction [i] would yield digital (57, 201, 340)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰¹, 340, F₄, 144) (dual of [340, 139, 145]-code), but
  - residual code [i] would yield OA(4⁵⁷, 195, S₄, 36), but
    - the linear programming bound shows that MÂ â‰¥ 494 490067 436667 455875 598203 818411 591542 653847 684514 372082 636620 702806 966272 / 22309 972171 916724 959618 830756 395146 773299 > 4⁵⁷ [i]

There is no (57, 204, 384)-net in base 4, because

1 times m-reduction [i] would yield (57, 203, 384)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 178 578538 299346 006593 291987 734909 832247 195845 784169 272780 116240 526553 639889 952496 736441 166550 100437 994873 658008 445669 158823 > 4²⁰³ [i]