MinT - Best Known (237−173, 237, s)-Nets in Base 4

Best Known (237−173, 237, s)-Nets in Base 4

Digital (64, 237, 66)-net over F₄, using

t-expansion [i] based on digital (49, 237, 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 (64, 237, 99)-net over F₄, using

t-expansion [i] based on digital (61, 237, 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 (64, 237, 332)-net over F₄, because

1 times m-reduction [i] would yield digital (64, 236, 332)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁶, 332, F₄, 172) (dual of [332, 96, 173]-code), but
  - residual code [i] would yield OA(4⁶⁴, 159, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 45270 819771 373261 134814 974786 535179 781633 219686 501979 035125 061347 877742 643588 123815 870907 832622 000802 701749 559691 837440 / 132 156573 038196 016753 662455 730323 092222 536469 928086 508825 109263 446128 015973 837189 > 4⁶⁴ [i]

There is no (64, 237, 424)-net in base 4, because

1 times m-reduction [i] would yield (64, 236, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13181 212775 882640 687993 803810 513352 302614 799209 253832 328884 924068 473022 555021 752206 238445 336331 145994 269085 489043 279614 273334 816313 527992 423764 > 4²³⁶ [i]