MinT - Best Known (61, 212, s)-Nets in Base 4

Best Known (61, 212, s)-Nets in Base 4

Digital (61, 212, 66)-net over F₄, using

t-expansion [i] based on digital (49, 212, 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 (61, 212, 99)-net over F₄, using

There is no digital (61, 212, 400)-net over F₄, because

3 times m-reduction [i] would yield digital (61, 209, 400)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁹, 400, F₄, 148) (dual of [400, 191, 149]-code), but
  - residual code [i] would yield OA(4⁶¹, 251, S₄, 37), but
    - the linear programming bound shows that MÂ â‰¥ 107 684832 476558 143648 589731 899624 769127 922671 724172 279174 574834 456723 456000 000000 / 20 050134 070696 232272 809918 735160 283681 235089 > 4⁶¹ [i]

There is no (61, 212, 415)-net in base 4, because

1 times m-reduction [i] would yield (61, 211, 415)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 019447 060020 000718 390251 002994 970478 069011 550280 336929 337808 453126 073834 753997 074809 551626 326622 881292 432097 166700 068914 830848 > 4²¹¹ [i]