MinT - Best Known (227−163, 227, s)-Nets in Base 4

Best Known (227−163, 227, s)-Nets in Base 4

Digital (64, 227, 66)-net over F₄, using

t-expansion [i] based on digital (49, 227, 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 (64, 227, 99)-net over F₄, using

t-expansion [i] based on digital (61, 227, 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 (64, 227, 385)-net over F₄, because

3 times m-reduction [i] would yield digital (64, 224, 385)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁴, 385, F₄, 160) (dual of [385, 161, 161]-code), but
  - residual code [i] would yield OA(4⁶⁴, 224, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 13361 740924 714558 183981 708538 240631 845789 107040 889033 084067 601409 088040 474771 456000 / 36 499780 543012 793452 506240 719222 345774 789003 > 4⁶⁴ [i]

There is no (64, 227, 431)-net in base 4, because

1 times m-reduction [i] would yield (64, 226, 431)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13489 102519 131091 211063 376841 676940 466785 682396 487467 963266 927728 466134 546471 000338 701040 956986 158464 547539 587956 644724 426388 511913 856440 > 4²²⁶ [i]