MinT - Best Known (229−163, 229, s)-Nets in Base 4

Best Known (229−163, 229, s)-Nets in Base 4

Digital (66, 229, 66)-net over F₄, using

t-expansion [i] based on digital (49, 229, 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 (66, 229, 99)-net over F₄, using

t-expansion [i] based on digital (61, 229, 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 (66, 229, 422)-net over F₄, because

3 times m-reduction [i] would yield digital (66, 226, 422)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁶, 422, F₄, 160) (dual of [422, 196, 161]-code), but
  - residual code [i] would yield OA(4⁶⁶, 261, S₄, 40), but
    - the linear programming bound shows that MÂ â‰¥ 327016 444997 139005 823489 073156 791701 824991 264795 866993 154478 474670 227349 271740 416000 / 57 698874 415078 925905 565190 919370 722090 693979 > 4⁶⁶ [i]

There is no (66, 229, 448)-net in base 4, because

1 times m-reduction [i] would yield (66, 228, 448)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 211522 055782 841248 735262 069629 443809 261294 782143 456372 230563 379010 241987 300502 162986 170951 870421 430692 001746 859827 277299 986645 086351 080150 > 4²²⁸ [i]