MinT - Best Known (68, 68+173, s)-Nets in Base 4

Best Known (68, 68+173, s)-Nets in Base 4

Digital (68, 241, 66)-net over F₄, using

t-expansion [i] based on digital (49, 241, 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, 241, 99)-net over F₄, using

t-expansion [i] based on digital (61, 241, 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, 241, 399)-net over F₄, because

1 times m-reduction [i] would yield digital (68, 240, 399)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁰, 399, F₄, 172) (dual of [399, 159, 173]-code), but
  - residual code [i] would yield OA(4⁶⁸, 226, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 25033 445810 985059 950340 011170 691662 989505 153485 270934 663957 957692 389934 318058 423546 481065 800499 200000 000000 / 283251 267996 901380 825721 338395 747776 993628 726657 913430 519555 536299 > 4⁶⁸ [i]

There is no (68, 241, 457)-net in base 4, because

1 times m-reduction [i] would yield (68, 240, 457)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 651209 105916 648276 346204 295166 755492 180825 322691 192852 516880 241751 438524 894297 604114 341406 106787 323389 286210 649492 884274 159416 919881 712629 520160 > 4²⁴⁰ [i]