MinT - Best Known (66, 66+175, s)-Nets in Base 4

Best Known (66, 66+175, s)-Nets in Base 4

Digital (66, 241, 66)-net over F₄, using

t-expansion [i] based on digital (49, 241, 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, 241, 99)-net over F₄, using

t-expansion [i] based on digital (61, 241, 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, 241, 364)-net over F₄, because

3 times m-reduction [i] would yield digital (66, 238, 364)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁸, 364, F₄, 172) (dual of [364, 126, 173]-code), but
  - residual code [i] would yield OA(4⁶⁶, 191, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 28 766535 420054 304848 397361 421684 986456 320614 942528 963784 752261 977481 718335 232020 438414 046131 126272 000000 / 5209 403279 129645 739501 754543 221181 850628 166933 877437 442223 389477 > 4⁶⁶ [i]

There is no (66, 241, 439)-net in base 4, because

1 times m-reduction [i] would yield (66, 240, 439)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 539970 243866 993779 102542 518931 578593 174596 676345 503318 290140 210157 956558 458607 132269 706561 685535 711504 344417 229492 354429 460022 259766 589031 278200 > 4²⁴⁰ [i]