MinT - Best Known (69, 252, s)-Nets in Base 4

Best Known (69, 252, s)-Nets in Base 4

Digital (69, 252, 66)-net over F₄, using

t-expansion [i] based on digital (49, 252, 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 (69, 252, 99)-net over F₄, using

t-expansion [i] based on digital (61, 252, 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 (69, 252, 376)-net over F₄, because

3 times m-reduction [i] would yield digital (69, 249, 376)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁹, 376, F₄, 180) (dual of [376, 127, 181]-code), but
  - residual code [i] would yield OA(4⁶⁹, 195, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 82 828010 425088 398135 365726 677314 464149 340113 774380 345347 664035 522400 039552 947596 743692 557981 180235 874304 000000 / 227 578582 237535 291818 752893 814619 068871 044126 923591 235904 712727 792773 > 4⁶⁹ [i]

There is no (69, 252, 458)-net in base 4, because

1 times m-reduction [i] would yield (69, 251, 458)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14 484646 198331 361667 056827 938982 657993 192134 074927 341719 009239 773331 352069 708567 741554 890262 468574 380753 926813 833347 764127 812679 909679 590456 177741 016000 > 4²⁵¹ [i]