MinT - Best Known (58, 208, s)-Nets in Base 4

Best Known (58, 208, s)-Nets in Base 4

Digital (58, 208, 66)-net over F₄, using

t-expansion [i] based on digital (49, 208, 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 (58, 208, 91)-net over F₄, using

t-expansion [i] based on digital (50, 208, 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 (58, 208, 342)-net over F₄, because

2 times m-reduction [i] would yield digital (58, 206, 342)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁶, 342, F₄, 148) (dual of [342, 136, 149]-code), but
  - residual code [i] would yield OA(4⁵⁸, 193, S₄, 37), but
    - the linear programming bound shows that MÂ â‰¥ 161285 735386 400830 134445 178003 287186 762218 409717 533224 067961 483465 588736 000000 / 1 868030 896794 654733 447451 649501 557791 288931 > 4⁵⁸ [i]

There is no (58, 208, 389)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 170315 259224 821412 221286 710444 305336 155538 531891 192890 514272 012295 331328 694708 673548 505822 423469 569624 064718 390893 642296 141504 > 4²⁰⁸ [i]