MinT - Best Known (226−167, 226, s)-Nets in Base 4

Best Known (226−167, 226, s)-Nets in Base 4

Digital (59, 226, 66)-net over F₄, using

t-expansion [i] based on digital (49, 226, 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 (59, 226, 91)-net over F₄, using

t-expansion [i] based on digital (50, 226, 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 (59, 226, 290)-net over F₄, because

3 times m-reduction [i] would yield digital (59, 223, 290)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²³, 290, F₄, 164) (dual of [290, 67, 165]-code), but
  - residual code [i] would yield OA(4⁵⁹, 125, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 6 098659 704705 486904 667441 402011 989050 548288 910427 912825 765842 283670 760248 415348 522730 039189 577335 282788 475369 967101 883338 375933 306396 897634 126855 141698 525692 111751 859853 590528 / 17 865783 539531 952003 937094 580988 623442 037272 256331 005047 247373 356459 142621 652981 139301 072409 181322 127802 625025 609129 027701 857126 610015 951793 > 4⁵⁹ [i]

There is no (59, 226, 389)-net in base 4, because

1 times m-reduction [i] would yield (59, 225, 389)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3466 944274 565295 706479 264273 478653 459859 761202 086219 025037 262633 098500 687956 445451 505166 236769 386281 595841 365090 901259 069304 274464 220000 > 4²²⁵ [i]