MinT - Best Known (58, 199, s)-Nets in Base 4

Best Known (58, 199, s)-Nets in Base 4

Digital (58, 199, 66)-net over F₄, using

t-expansion [i] based on digital (49, 199, 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, 199, 91)-net over F₄, using

t-expansion [i] based on digital (50, 199, 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, 199, 391)-net over F₄, because

1 times m-reduction [i] would yield digital (58, 198, 391)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁸, 391, F₄, 140) (dual of [391, 193, 141]-code), but
  - residual code [i] would yield OA(4⁵⁸, 250, S₄, 35), but
    - the linear programming bound shows that MÂ â‰¥ 1 169022 325481 903034 671724 153754 218504 051580 462823 775601 886542 980534 239232 / 13 842163 950449 124319 768889 299435 690739 > 4⁵⁸ [i]

There is no (58, 199, 398)-net in base 4, because

1 times m-reduction [i] would yield (58, 198, 398)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181864 060264 355758 441323 999235 417238 573849 006761 077361 934035 527458 504087 008321 912404 871428 851686 884214 183212 605091 862456 > 4¹⁹⁸ [i]