MinT - Best Known (218−162, 218, s)-Nets in Base 4

Best Known (218−162, 218, s)-Nets in Base 4

Digital (56, 218, 66)-net over F₄, using

t-expansion [i] based on digital (49, 218, 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, 218, 91)-net over F₄, using

t-expansion [i] based on digital (50, 218, 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, 218, 259)-net over F₄, because

2 times m-reduction [i] would yield digital (56, 216, 259)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁶, 259, F₄, 160) (dual of [259, 43, 161]-code), but
  - residual code [i] would yield OA(4⁵⁶, 98, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 108 858270 389765 832134 899494 245555 240896 445146 215984 684712 894959 792949 029119 197184 / 19965 002432 218887 804054 871233 298540 608350 321175 > 4⁵⁶ [i]

There is no (56, 218, 368)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200765 993064 849353 223622 144469 148712 739311 421956 136151 692404 052951 081179 443612 561299 060951 246708 034440 376420 930806 503519 118092 681270 > 4²¹⁸ [i]