MinT - Best Known (230−163, 230, s)-Nets in Base 4

Best Known (230−163, 230, s)-Nets in Base 4

Digital (67, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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, 230, 99)-net over F₄, using

t-expansion [i] based on digital (61, 230, 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, 230, 447)-net over F₄, because

3 times m-reduction [i] would yield digital (67, 227, 447)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁷, 447, F₄, 160) (dual of [447, 220, 161]-code), but
  - residual code [i] would yield OA(4⁶⁷, 286, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 2248 462530 540235 331681 045610 925265 548739 257567 179323 047117 943462 838604 815338 045440 000000 / 99794 086939 136362 792787 734261 406363 880711 755149 > 4⁶⁷ [i]

There is no (67, 230, 456)-net in base 4, because

1 times m-reduction [i] would yield (67, 229, 456)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 748139 763192 437273 023422 161082 737646 835783 586703 819149 341788 611563 121559 270574 104677 150033 578328 420730 123425 677840 855295 614069 449793 390456 > 4²²⁹ [i]