MinT - Best Known (67, 67+175, s)-Nets in Base 4

Best Known (67, 67+175, s)-Nets in Base 4

Digital (67, 242, 66)-net over F₄, using

t-expansion [i] based on digital (49, 242, 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 (67, 242, 99)-net over F₄, using

t-expansion [i] based on digital (61, 242, 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 (67, 242, 380)-net over F₄, because

3 times m-reduction [i] would yield digital (67, 239, 380)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁹, 380, F₄, 172) (dual of [380, 141, 173]-code), but
  - residual code [i] would yield OA(4⁶⁷, 207, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 28014 561764 993307 241664 052911 871857 702310 880709 902206 525389 902012 241710 106857 807269 754280 270102 528000 / 1 234731 917948 789567 037359 176277 481011 072922 589066 284983 746001 > 4⁶⁷ [i]

There is no (67, 242, 447)-net in base 4, because

1 times m-reduction [i] would yield (67, 241, 447)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 931476 613696 215026 809292 416429 770365 386514 501835 739244 062899 891580 769084 222349 038826 667149 271606 624050 078297 253375 889553 159767 971102 205857 823400 > 4²⁴¹ [i]