MinT - Best Known (64, 245, s)-Nets in Base 4

Best Known (64, 245, s)-Nets in Base 4

Digital (64, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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, 245, 300)-net over F₄, because

1 times m-reduction [i] would yield digital (64, 244, 300)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁴, 300, F₄, 180) (dual of [300, 56, 181]-code), but
  - residual code [i] would yield OA(4⁶⁴, 119, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 86281 259132 858652 351318 621152 193776 169011 314302 048568 574075 512291 106476 673632 920228 434592 062075 337852 335348 631534 055263 738553 834940 174592 676821 962753 440810 532864 / 242 561963 734980 780383 061022 729662 792650 160301 201736 163590 343831 386934 538599 980695 527882 064362 246641 801684 355864 415256 969895 > 4⁶⁴ [i]

There is no (64, 245, 420)-net in base 4, because

1 times m-reduction [i] would yield (64, 244, 420)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 844 150664 819872 865753 252513 800307 792192 237098 106372 168594 548412 960369 160790 311426 637189 601600 360870 648988 058366 838949 928015 110491 125641 884774 966536 > 4²⁴⁴ [i]