MinT - Best Known (71, 257, s)-Nets in Base 4

Best Known (71, 257, s)-Nets in Base 4

Digital (71, 257, 66)-net over F₄, using

t-expansion [i] based on digital (49, 257, 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, 257, 105)-net over F₄, using

t-expansion [i] based on digital (70, 257, 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, 257, 391)-net over F₄, because

2 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, 257, 472)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 63680 021766 800990 310472 165099 367525 748382 176148 417687 300479 262233 612692 652594 098050 593208 404041 597295 308782 900025 579148 990898 591324 288304 473844 339478 149776 > 4²⁵⁷ [i]