MinT - Best Known (224−162, 224, s)-Nets in Base 4

Best Known (224−162, 224, s)-Nets in Base 4

Digital (62, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 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 (62, 224, 99)-net over F₄, using

t-expansion [i] based on digital (61, 224, 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 (62, 224, 352)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 222, 352)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²², 352, F₄, 160) (dual of [352, 130, 161]-code), but
  - residual code [i] would yield OA(4⁶², 191, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 6 721352 642212 013290 164202 930663 188729 706683 395933 433874 909615 380474 388774 465686 228242 006016 / 300843 081103 152389 559805 379819 997539 174337 778277 134589 > 4⁶² [i]

There is no (62, 224, 414)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 779 923772 399501 257536 365694 847592 472914 722743 553711 138056 535870 773835 701612 578818 637842 194144 203799 960704 194747 775947 811141 010572 713210 > 4²²⁴ [i]