MinT - Best Known (71, 71+189, s)-Nets in Base 4

Best Known (71, 71+189, s)-Nets in Base 4

Digital (71, 260, 66)-net over F₄, using

t-expansion [i] based on digital (49, 260, 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 (71, 260, 105)-net over F₄, using

t-expansion [i] based on digital (70, 260, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

There is no digital (71, 260, 381)-net over F₄, because

1 times m-reduction [i] would yield digital (71, 259, 381)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁹, 381, F₄, 188) (dual of [381, 122, 189]-code), but
  - residual code [i] would yield OA(4⁷¹, 192, S₄, 47), but
    - 1 times truncation [i] would yield OA(4⁷⁰, 191, S₄, 46), but
      - the linear programming bound shows that MÂ â‰¥ 30669 687295 767807 163249 547418 721993 874758 654208 012151 612633 672044 408069 277192 616104 296206 434138 576650 240000 / 20720 048409 770029 043765 568578 931955 503324 140138 849174 037831 490849 > 4⁷⁰ [i]

There is no (71, 260, 470)-net in base 4, because

1 times m-reduction [i] would yield (71, 259, 470)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 891897 294710 102498 825888 006652 921579 967529 379058 033011 138631 023443 350684 655003 618258 395353 276864 383252 217757 622931 772219 004523 684609 570228 261674 486592 384064 > 4²⁵⁹ [i]