MinT - Best Known (63, 63+162, s)-Nets in Base 4

Best Known (63, 63+162, s)-Nets in Base 4

Digital (63, 225, 66)-net over F₄, using

t-expansion [i] based on digital (49, 225, 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 (63, 225, 99)-net over F₄, using

t-expansion [i] based on digital (61, 225, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

There is no digital (63, 225, 369)-net over F₄, because

2 times m-reduction [i] would yield digital (63, 223, 369)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²³, 369, F₄, 160) (dual of [369, 146, 161]-code), but
  - residual code [i] would yield OA(4⁶³, 208, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 5 853210 937914 115965 830373 633983 054380 602394 595992 687313 196476 039443 785703 042842 624000 / 66248 338389 800632 875754 408934 650630 828021 253383 > 4⁶³ [i]

There is no (63, 225, 422)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3020 668282 556385 082616 063072 597084 883558 102406 972737 448124 091205 134243 397912 184663 421787 671040 823771 000446 593800 758631 360569 858743 077956 > 4²²⁵ [i]