MinT - Best Known (245−178, 245, s)-Nets in Base 4

Best Known (245−178, 245, s)-Nets in Base 4

Digital (67, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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, 245, 362)-net over F₄, because

2 times m-reduction [i] would yield digital (67, 243, 362)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴³, 362, F₄, 176) (dual of [362, 119, 177]-code), but
  - residual code [i] would yield OA(4⁶⁷, 185, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 27 922736 614479 963563 498030 039555 280180 497392 114764 937434 686895 163379 799248 329552 391127 539087 989670 936576 / 1212 198916 755108 948361 633672 026124 789528 916769 954260 490443 945723 > 4⁶⁷ [i]

There is no (67, 245, 444)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3278 424737 262322 591866 494083 774736 120825 367484 644165 883183 151953 920163 538523 839121 750028 742585 100398 393510 588288 009249 834724 563574 489864 824629 631530 > 4²⁴⁵ [i]