MinT - Best Known (55, 209, s)-Nets in Base 4

Best Known (55, 209, s)-Nets in Base 4

Digital (55, 209, 66)-net over F₄, using

t-expansion [i] based on digital (49, 209, 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 (55, 209, 91)-net over F₄, using

t-expansion [i] based on digital (50, 209, 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 (55, 209, 278)-net over F₄, because

2 times m-reduction [i] would yield digital (55, 207, 278)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁷, 278, F₄, 152) (dual of [278, 71, 153]-code), but
  - residual code [i] would yield OA(4⁵⁵, 125, S₄, 38), but
    - the linear programming bound shows that MÂ â‰¥ 319 069835 724865 932418 074123 003651 341257 341197 442579 627763 661845 151026 883688 745199 999424 563055 993464 060641 280000 / 235791 539716 773839 952561 097991 148290 075568 550579 267395 341876 457603 747103 745171 > 4⁵⁵ [i]

There is no (55, 209, 364)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 805200 278938 896038 795575 040908 101453 200720 304980 610592 697885 458886 179925 367636 778099 569697 715261 513626 433074 095480 511539 210280 > 4²⁰⁹ [i]