MinT - Best Known (68, 68+183, s)-Nets in Base 4

Best Known (68, 68+183, s)-Nets in Base 4

Digital (68, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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 (68, 251, 99)-net over F₄, using

t-expansion [i] based on digital (61, 251, 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 (68, 251, 361)-net over F₄, because

3 times m-reduction [i] would yield digital (68, 248, 361)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁸, 361, F₄, 180) (dual of [361, 113, 181]-code), but
  - residual code [i] would yield OA(4⁶⁸, 180, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 328494 790735 001198 305199 602445 532392 471737 863161 499929 581656 031112 774235 621273 901469 693549 435944 960000 000000 / 3 747617 146367 023709 191641 153229 380578 240235 288823 915002 005290 401803 > 4⁶⁸ [i]

There is no (68, 251, 450)-net in base 4, because

1 times m-reduction [i] would yield (68, 250, 450)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 590213 184119 779345 194160 674216 650098 014067 079553 757418 822826 785892 482271 876582 991660 656710 736322 774627 171039 242318 852036 567118 369727 057339 870514 696640 > 4²⁵⁰ [i]