MinT - Best Known (62, 240, s)-Nets in Base 4

Best Known (62, 240, s)-Nets in Base 4

Digital (62, 240, 66)-net over F₄, using

t-expansion [i] based on digital (49, 240, 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 (62, 240, 99)-net over F₄, using

t-expansion [i] based on digital (61, 240, 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 (62, 240, 284)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 238, 284)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁸, 284, F₄, 176) (dual of [284, 46, 177]-code), but
  - residual code [i] would yield OA(4⁶², 107, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 17622 602942 558177 133437 108590 576398 639747 070941 352929 752748 829629 227505 655642 080394 898835 505152 / 826 744207 227851 738915 816443 333621 359995 803052 054368 328125 > 4⁶² [i]

There is no (62, 240, 406)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 339631 801306 184299 986930 243431 822661 409963 990688 875555 139348 377308 303127 794830 138219 702814 340278 656898 188051 862012 296815 478495 174952 327591 559120 > 4²⁴⁰ [i]