MinT - Best Known (247−183, 247, s)-Nets in Base 4

Best Known (247−183, 247, s)-Nets in Base 4

Digital (64, 247, 66)-net over F₄, using

t-expansion [i] based on digital (49, 247, 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, 247, 99)-net over F₄, using

t-expansion [i] based on digital (61, 247, 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, 247, 300)-net over F₄, because

3 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, 247, 419)-net in base 4, because

1 times m-reduction [i] would yield (64, 246, 419)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13018 662606 900284 079875 950028 577985 257952 708263 392320 058632 273564 939249 629198 519767 153840 078947 817237 419152 568984 606453 719921 234213 752624 872139 915488 > 4²⁴⁶ [i]