MinT - Best Known (194−138, 194, s)-Nets in Base 4

Best Known (194−138, 194, s)-Nets in Base 4

Digital (56, 194, 66)-net over F₄, using

t-expansion [i] based on digital (49, 194, 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, 194, 91)-net over F₄, using

t-expansion [i] based on digital (50, 194, 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, 194, 374)-net over F₄, because

2 times m-reduction [i] would yield digital (56, 192, 374)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹², 374, F₄, 136) (dual of [374, 182, 137]-code), but
  - residual code [i] would yield OA(4⁵⁶, 237, S₄, 34), but
    - the linear programming bound shows that MÂ â‰¥ 227 819482 170490 726334 286831 192651 607943 130369 749048 513507 843112 960000 000000 / 43777 691396 586787 052635 611770 906797 710989 > 4⁵⁶ [i]

There is no (56, 194, 382)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 685 168564 230783 213901 431432 075489 910368 639202 505936 021580 244551 821417 310536 732725 775393 318618 575998 069035 782134 458107 > 4¹⁹⁴ [i]