MinT - Best Known (241−177, 241, s)-Nets in Base 4

Best Known (241−177, 241, s)-Nets in Base 4

Digital (64, 241, 66)-net over F₄, using

t-expansion [i] based on digital (49, 241, 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 (64, 241, 99)-net over F₄, using

t-expansion [i] based on digital (61, 241, 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 (64, 241, 317)-net over F₄, because

1 times m-reduction [i] would yield digital (64, 240, 317)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁰, 317, F₄, 176) (dual of [317, 77, 177]-code), but
  - residual code [i] would yield OA(4⁶⁴, 140, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 6 055971 130979 717035 237533 763765 896035 104411 078322 823391 498324 273826 156962 484218 588303 215205 346683 113225 234756 746893 442044 988412 906254 866058 803915 928912 265216 000000 / 17465 630993 245964 374557 288355 588936 264526 909741 890553 067641 295523 141418 950639 065672 191422 532782 852686 274622 640710 777054 531879 > 4⁶⁴ [i]

There is no (64, 241, 422)-net in base 4, because

1 times m-reduction [i] would yield (64, 240, 422)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 420318 426758 635266 849754 454386 078200 846986 241892 805467 944062 423578 430168 002951 778353 760217 481841 896449 884366 346279 636767 976594 419779 697598 189064 > 4²⁴⁰ [i]