MinT - Best Known (70, 252, s)-Nets in Base 4

Best Known (70, 252, s)-Nets in Base 4

Digital (70, 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 (70, 252, 105)-net over F₄, using

There is no digital (70, 252, 394)-net over F₄, because

2 times m-reduction [i] would yield digital (70, 250, 394)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁰, 394, F₄, 180) (dual of [394, 144, 181]-code), but
  - residual code [i] would yield OA(4⁷⁰, 213, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 10138 160863 655691 762503 861603 731431 648994 530508 429756 680407 696662 337185 130982 211720 965164 892160 / 6819 841021 187779 545947 587667 623913 181856 482312 426343 > 4⁷⁰ [i]

There is no (70, 252, 466)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 57 205214 733067 065835 257247 378536 388991 828470 763385 163486 843481 363315 370524 062227 085123 668243 910359 762082 949483 612876 554501 242543 835813 788207 711496 296256 > 4²⁵² [i]