MinT - Best Known (74, 256, s)-Nets in Base 4

Best Known (74, 256, s)-Nets in Base 4

Digital (74, 256, 104)-net over F₄, using

t-expansion [i] based on digital (73, 256, 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 (74, 256, 112)-net over F₄, using

t-expansion [i] based on digital (73, 256, 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 (74, 256, 467)-net over F₄, because

2 times m-reduction [i] would yield digital (74, 254, 467)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 467, F₄, 180) (dual of [467, 213, 181]-code), but
  - residual code [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 (74, 256, 500)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 15692 362447 398367 800924 362399 493906 185389 785815 178538 728571 185092 475961 357233 380612 726482 825869 175380 108317 672981 825933 832988 594880 152800 806781 909391 588756 > 4²⁵⁶ [i]