MinT - Best Known (65, 65+186, s)-Nets in Base 4

Best Known (65, 65+186, s)-Nets in Base 4

Digital (65, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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 (65, 251, 99)-net over F₄, using

t-expansion [i] based on digital (61, 251, 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 (65, 251, 298)-net over F₄, because

2 times m-reduction [i] would yield digital (65, 249, 298)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁹, 298, F₄, 184) (dual of [298, 49, 185]-code), but
  - residual code [i] would yield OA(4⁶⁵, 113, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 145 402369 805359 268193 209685 952643 209942 587568 332064 303419 732807 154505 932977 868145 129044 024267 389501 535040 862760 730624 / 100423 807776 117676 644806 780906 988983 236254 180840 898357 650546 636091 023031 034405 > 4⁶⁵ [i]

There is no (65, 251, 425)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 636844 987413 756773 372537 549395 524449 948059 799064 100986 684984 496936 510430 269379 856452 931345 090274 386292 974212 402703 671921 144003 478503 318487 227064 540288 > 4²⁵¹ [i]