MinT - Best Known (71, 256, s)-Nets in Base 4

Best Known (71, 256, s)-Nets in Base 4

Digital (71, 256, 66)-net over F₄, using

t-expansion [i] based on digital (49, 256, 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 (71, 256, 105)-net over F₄, using

t-expansion [i] based on digital (70, 256, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

There is no digital (71, 256, 391)-net over F₄, because

1 times m-reduction [i] would yield digital (71, 255, 391)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁵, 391, F₄, 184) (dual of [391, 136, 185]-code), but
  - residual code [i] would yield OA(4⁷¹, 206, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 12190 666552 860549 875030 280693 949215 033597 089167 787297 540990 306499 998443 349891 008938 804128 810849 723006 100908 802048 / 2095 122091 727468 263392 248874 603589 540011 988045 238608 933987 894642 253863 > 4⁷¹ [i]

There is no (71, 256, 473)-net in base 4, because

1 times m-reduction [i] would yield (71, 255, 473)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3776 375302 168869 743853 712820 407334 849599 352816 491107 668600 760066 343101 536939 315543 235001 627677 922933 899299 387723 167514 089711 983783 853560 676367 323949 857696 > 4²⁵⁵ [i]