MinT - Best Known (227−166, 227, s)-Nets in Base 4

Best Known (227−166, 227, s)-Nets in Base 4

Digital (61, 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 (61, 227, 99)-net over F₄, using

There is no digital (61, 227, 320)-net over F₄, because

2 times m-reduction [i] would yield digital (61, 225, 320)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁵, 320, F₄, 164) (dual of [320, 95, 165]-code), but
  - residual code [i] would yield OA(4⁶¹, 155, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 35 162266 348965 864492 465363 741528 241521 515995 482555 223491 846236 883313 207330 736210 116983 129154 519040 / 6 178619 757792 266774 478203 941876 542733 832278 327110 913669 619231 > 4⁶¹ [i]

There is no (61, 227, 404)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52500 719087 102455 387342 841227 595162 671401 055054 096946 990812 774704 075468 767908 613754 552297 280066 181931 839861 115722 003496 546500 485589 351494 > 4²²⁷ [i]