MinT - Best Known (254−183, 254, s)-Nets in Base 4

Best Known (254−183, 254, s)-Nets in Base 4

Digital (71, 254, 66)-net over F₄, using

t-expansion [i] based on digital (49, 254, 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 (71, 254, 105)-net over F₄, using

t-expansion [i] based on digital (70, 254, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

There is no digital (71, 254, 410)-net over F₄, because

3 times m-reduction [i] would yield digital (71, 251, 410)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵¹, 410, F₄, 180) (dual of [410, 159, 181]-code), but
  - residual code [i] would yield OA(4⁷¹, 229, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 877 912702 924905 625680 913013 034326 550228 995916 587809 875578 704494 502776 185113 709858 393246 285168 640000 / 155 079140 374101 434134 168939 696571 367080 998053 409783 569671 > 4⁷¹ [i]

There is no (71, 254, 474)-net in base 4, because

1 times m-reduction [i] would yield (71, 253, 474)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 221 301973 105587 822471 229154 721119 982733 190093 084467 902071 676350 360006 007253 910374 507580 050584 999388 690879 404431 466475 910032 194517 482755 372025 798687 548656 > 4²⁵³ [i]