MinT - Best Known (225−167, 225, s)-Nets in Base 4

Best Known (225−167, 225, s)-Nets in Base 4

Digital (58, 225, 66)-net over F₄, using

t-expansion [i] based on digital (49, 225, 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 (58, 225, 91)-net over F₄, using

t-expansion [i] based on digital (50, 225, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

There is no digital (58, 225, 275)-net over F₄, because

3 times m-reduction [i] would yield digital (58, 222, 275)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²², 275, F₄, 164) (dual of [275, 53, 165]-code), but
  - residual code [i] would yield OA(4⁵⁸, 110, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 1232 155961 216230 168086 977678 056475 191089 409710 805770 829825 897828 417067 677196 009558 961106 027628 461497 995772 222254 044023 160230 949557 990006 504845 840607 997118 369373 847851 273455 403008 / 13446 810158 281327 760184 995983 919490 800441 646762 168115 151101 794360 608708 987818 154852 821096 884734 101312 394526 499211 581201 357552 019470 400051 102575 > 4⁵⁸ [i]

There is no (58, 225, 381)-net in base 4, because

1 times m-reduction [i] would yield (58, 224, 381)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 783 608161 753905 663759 659235 762498 392786 098932 291934 706945 039735 447532 988950 534097 672487 653102 851463 728294 117217 137420 141856 353885 546772 > 4²²⁴ [i]