MinT - Best Known (259−187, 259, s)-Nets in Base 4

Best Known (259−187, 259, s)-Nets in Base 4

Digital (72, 259, 66)-net over F₄, using

t-expansion [i] based on digital (49, 259, 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 (72, 259, 105)-net over F₄, using

t-expansion [i] based on digital (70, 259, 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 (72, 259, 408)-net over F₄, because

3 times m-reduction [i] would yield digital (72, 256, 408)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁶, 408, F₄, 184) (dual of [408, 152, 185]-code), but
  - residual code [i] would yield OA(4⁷², 223, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 360 417143 929919 650649 839526 859205 286331 400168 894487 479850 253672 313831 388847 004885 511580 359694 158397 440000 000000 / 15 992885 341550 871668 440406 576991 261654 953271 872698 021682 171330 664199 > 4⁷² [i]

There is no (72, 259, 480)-net in base 4, because

1 times m-reduction [i] would yield (72, 258, 480)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 249035 780013 517287 376998 832914 955719 608834 103106 561588 115996 339456 483297 128208 447167 126708 982321 960180 408567 805167 206162 174404 689966 307465 131254 930081 340749 > 4²⁵⁸ [i]