MinT - Best Known (210−154, 210, s)-Nets in Base 4

Best Known (210−154, 210, s)-Nets in Base 4

Digital (56, 210, 66)-net over F₄, using

t-expansion [i] based on digital (49, 210, 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 (56, 210, 91)-net over F₄, using

t-expansion [i] based on digital (50, 210, 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 (56, 210, 293)-net over F₄, because

2 times m-reduction [i] would yield digital (56, 208, 293)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁸, 293, F₄, 152) (dual of [293, 85, 153]-code), but
  - residual code [i] would yield OA(4⁵⁶, 140, S₄, 38), but
    - the linear programming bound shows that MÂ â‰¥ 33224 510089 193434 121103 982150 072093 135568 373830 671910 070349 424562 458616 197168 496640 000000 / 5 929310 526700 508468 903034 465148 531898 302680 222550 630653 > 4⁵⁶ [i]

There is no (56, 210, 371)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 864970 970784 042241 896263 923409 075624 528403 297388 575970 854246 791202 056627 334935 926555 853946 160069 559342 569457 175768 670955 967416 > 4²¹⁰ [i]