MinT - Best Known (58, 213, s)-Nets in Base 4

Best Known (58, 213, s)-Nets in Base 4

Digital (58, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 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, 213, 91)-net over F₄, using

t-expansion [i] based on digital (50, 213, 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, 213, 322)-net over F₄, because

3 times m-reduction [i] would yield digital (58, 210, 322)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁰, 322, F₄, 152) (dual of [322, 112, 153]-code), but
  - residual code [i] would yield OA(4⁵⁸, 169, S₄, 38), but
    - the linear programming bound shows that MÂ â‰¥ 6 392121 980617 372713 740932 173054 416763 440978 777125 251070 801955 756485 046278 553600 000000 / 72 300057 426700 433200 861870 971600 390242 400928 758347 > 4⁵⁸ [i]

There is no (58, 213, 387)-net in base 4, because

1 times m-reduction [i] would yield (58, 212, 387)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48 250712 385890 501428 579776 383069 504733 252622 399070 395448 078239 692554 419244 207306 695264 706151 915852 088610 276445 855231 153158 363200 > 4²¹² [i]