MinT - Best Known (234−167, 234, s)-Nets in Base 4

Best Known (234−167, 234, s)-Nets in Base 4

Digital (67, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 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 (67, 234, 99)-net over F₄, using

t-expansion [i] based on digital (61, 234, 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 (67, 234, 422)-net over F₄, because

3 times m-reduction [i] would yield digital (67, 231, 422)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³¹, 422, F₄, 164) (dual of [422, 191, 165]-code), but
  - residual code [i] would yield OA(4⁶⁷, 257, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 188282 701092 137183 285392 695738 133645 870848 581784 362420 528688 778251 423906 057085 255680 / 8 301637 427320 826521 830664 836391 530003 316903 > 4⁶⁷ [i]

There is no (67, 234, 453)-net in base 4, because

1 times m-reduction [i] would yield (67, 233, 453)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 660096 912400 317766 309503 977141 160529 082697 023017 431772 426616 760565 200493 774740 242496 761442 468093 539645 783846 879805 324194 052486 607560 759520 > 4²³³ [i]