MinT - Best Known (260−185, 260, s)-Nets in Base 4

Best Known (260−185, 260, s)-Nets in Base 4

Digital (75, 260, 104)-net over F₄, using

t-expansion [i] based on digital (73, 260, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (75, 260, 112)-net over F₄, using

t-expansion [i] based on digital (73, 260, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

There is no digital (75, 260, 472)-net over F₄, because

1 times m-reduction [i] would yield digital (75, 259, 472)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁹, 472, F₄, 184) (dual of [472, 213, 185]-code), but
  - residual code [i] would yield OA(4⁷⁵, 287, S₄, 46), but
    - 1 times truncation [i] would yield OA(4⁷⁴, 286, S₄, 45), but
      - the linear programming bound shows that MÂ â‰¥ 22 769120 707181 828041 954378 161822 084052 954335 244294 016427 586097 147930 001332 191588 780049 100806 553600 / 63648 956571 718794 371382 905612 240945 674126 728108 353777 > 4⁷⁴ [i]

There is no (75, 260, 506)-net in base 4, because

1 times m-reduction [i] would yield (75, 259, 506)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 867798 733369 160145 723612 638379 238265 623707 717532 569723 676714 503301 554676 289108 817435 088248 005224 268813 358270 574337 045421 601129 861734 170805 106575 142241 124952 > 4²⁵⁹ [i]