MinT - Best Known (242−178, 242, s)-Nets in Base 4

Best Known (242−178, 242, s)-Nets in Base 4

Digital (64, 242, 66)-net over F₄, using

t-expansion [i] based on digital (49, 242, 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, 242, 99)-net over F₄, using

t-expansion [i] based on digital (61, 242, 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, 242, 317)-net over F₄, because

2 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, 242, 421)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54 067575 515276 113652 430538 739486 678468 906427 152713 960736 052117 293513 083755 058168 271138 433035 789858 515669 167426 290252 686515 841520 913026 599596 136768 > 4²⁴² [i]