MinT - Best Known (69, 69+179, s)-Nets in Base 4

Best Known (69, 69+179, s)-Nets in Base 4

Digital (69, 248, 66)-net over F₄, using

t-expansion [i] based on digital (49, 248, 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, 248, 99)-net over F₄, using

t-expansion [i] based on digital (61, 248, 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, 248, 395)-net over F₄, because

3 times m-reduction [i] would yield digital (69, 245, 395)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁵, 395, F₄, 176) (dual of [395, 150, 177]-code), but
  - residual code [i] would yield OA(4⁶⁹, 218, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 109109 147274 768315 990931 948474 366203 066087 657712 317992 394776 862369 613323 594430 467474 878725 704856 371200 / 292745 629447 360691 756384 897898 834188 795527 737754 030331 320673 > 4⁶⁹ [i]

There is no (69, 248, 461)-net in base 4, because

1 times m-reduction [i] would yield (69, 247, 461)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 60504 347907 806502 464585 695550 257220 559786 623746 333605 045504 037315 070847 237329 316718 654032 025363 492076 501989 629603 136369 151748 717643 266529 559403 579136 > 4²⁴⁷ [i]