MinT - Best Known (68, 68+165, s)-Nets in Base 4

Best Known (68, 68+165, s)-Nets in Base 4

Digital (68, 233, 66)-net over F₄, using

t-expansion [i] based on digital (49, 233, 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 (68, 233, 99)-net over F₄, using

t-expansion [i] based on digital (61, 233, 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 (68, 233, 443)-net over F₄, because

1 times m-reduction [i] would yield digital (68, 232, 443)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³², 443, F₄, 164) (dual of [443, 211, 165]-code), but
  - residual code [i] would yield OA(4⁶⁸, 278, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 9751 854148 120896 570793 936092 472808 488071 901083 469159 873923 045115 979366 400000 000000 000000 / 106042 296287 604630 552636 245594 756356 027986 690833 > 4⁶⁸ [i]

There is no (68, 233, 463)-net in base 4, because

1 times m-reduction [i] would yield (68, 232, 463)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48 272388 828640 196956 362389 273140 223121 892184 703227 352706 962480 873345 296892 595303 246711 564894 794172 563056 754211 104998 395529 857615 222472 374840 > 4²³² [i]