MinT - Best Known (251−179, 251, s)-Nets in Base 4

Best Known (251−179, 251, s)-Nets in Base 4

Digital (72, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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 (72, 251, 105)-net over F₄, using

t-expansion [i] based on digital (70, 251, 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 (72, 251, 447)-net over F₄, because

3 times m-reduction [i] would yield digital (72, 248, 447)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁸, 447, F₄, 176) (dual of [447, 199, 177]-code), but
  - residual code [i] would yield OA(4⁷², 270, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 786435 471178 717771 595724 051341 292858 017461 789380 078325 091944 259711 116987 003218 742476 800000 000000 / 34141 957272 513139 588342 951169 793634 345796 839121 212019 > 4⁷² [i]

There is no (72, 251, 486)-net in base 4, because

1 times m-reduction [i] would yield (72, 250, 486)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 720932 680378 735408 075822 229969 604520 050154 446276 113660 021746 678807 226582 687835 746837 789649 821629 007819 420908 643216 663342 809589 142126 220696 502036 362108 > 4²⁵⁰ [i]