MinT - Best Known (69, 69+187, s)-Nets in Base 4

Best Known (69, 69+187, s)-Nets in Base 4

Digital (69, 256, 66)-net over F₄, using

t-expansion [i] based on digital (49, 256, 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 (69, 256, 99)-net over F₄, using

t-expansion [i] based on digital (61, 256, 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 (69, 256, 360)-net over F₄, because

3 times m-reduction [i] would yield digital (69, 253, 360)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵³, 360, F₄, 184) (dual of [360, 107, 185]-code), but
  - residual code [i] would yield OA(4⁶⁹, 175, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 672974 508783 713085 919446 392721 434812 374063 221813 152066 788676 290147 045574 983959 198536 005103 820640 027801 847804 723200 000000 / 1 829712 211075 097010 907956 226624 270704 376891 436886 295247 592787 694007 310519 329949 > 4⁶⁹ [i]

There is no (69, 256, 456)-net in base 4, because

1 times m-reduction [i] would yield (69, 255, 456)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3912 293643 495011 180315 995926 455644 194996 765064 692658 864838 681337 820740 915091 213048 958184 856854 532835 613089 743327 703398 432027 780327 091340 274831 253049 822544 > 4²⁵⁵ [i]