MinT - Best Known (226−166, 226, s)-Nets in Base 4

Best Known (226−166, 226, s)-Nets in Base 4

Digital (60, 226, 66)-net over F₄, using

t-expansion [i] based on digital (49, 226, 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 (60, 226, 91)-net over F₄, using

t-expansion [i] based on digital (50, 226, 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 (60, 226, 305)-net over F₄, because

2 times m-reduction [i] would yield digital (60, 224, 305)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁴, 305, F₄, 164) (dual of [305, 81, 165]-code), but
  - residual code [i] would yield OA(4⁶⁰, 140, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 879403 484026 313933 129107 893130 654952 488416 854359 107391 996054 021035 128392 974988 409099 175067 144881 999812 150052 449544 895148 851200 / 619301 085252 374849 001693 023953 651889 253595 081048 920563 530321 558861 109148 317743 673454 318139 > 4⁶⁰ [i]

There is no (60, 226, 396)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12462 177345 175139 116393 749452 403288 505007 071740 921141 600022 776448 127426 047334 169813 372409 827411 094791 051188 123557 245669 428613 577185 653176 > 4²²⁶ [i]