MinT - Best Known (67, 253, s)-Nets in Base 4

Best Known (67, 253, s)-Nets in Base 4

Digital (67, 253, 66)-net over F₄, using

t-expansion [i] based on digital (49, 253, 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, 253, 99)-net over F₄, using

t-expansion [i] based on digital (61, 253, 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, 253, 330)-net over F₄, because

2 times m-reduction [i] would yield digital (67, 251, 330)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵¹, 330, F₄, 184) (dual of [330, 79, 185]-code), but
  - residual code [i] would yield OA(4⁶⁷, 145, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 75449 327415 575534 134401 312271 358364 202586 988312 764269 677821 187597 326851 676773 310740 281541 900157 618938 630703 879340 724024 839258 891498 301173 934871 113793 698587 844036 163230 383972 563369 936814 080000 / 3 442382 810781 772421 076115 876775 468779 327697 682460 018151 373513 549713 415301 402628 670590 013076 362535 108026 742348 698089 724569 107664 952341 600266 478360 191877 > 4⁶⁷ [i]

There is no (67, 253, 440)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 220 230535 977764 796474 026361 343184 621499 345480 103314 400884 777773 085587 920872 420991 323997 434771 491222 664466 136841 237760 634771 855153 466760 179012 788549 433880 > 4²⁵³ [i]