MinT - Best Known (63, 229, s)-Nets in Base 4

Best Known (63, 229, s)-Nets in Base 4

Digital (63, 229, 66)-net over F₄, using

t-expansion [i] based on digital (49, 229, 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 (63, 229, 99)-net over F₄, using

t-expansion [i] based on digital (61, 229, 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 (63, 229, 350)-net over F₄, because

2 times m-reduction [i] would yield digital (63, 227, 350)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁷, 350, F₄, 164) (dual of [350, 123, 165]-code), but
  - residual code [i] would yield OA(4⁶³, 185, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 61 899957 150953 106035 135074 637879 046641 791483 715273 894502 688876 097802 046438 481549 501189 324800 / 720746 857989 042001 933213 860469 455977 050620 069897 655507 > 4⁶³ [i]

There is no (63, 229, 420)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 865655 047635 322824 696006 959109 688483 632183 159736 809583 234790 400797 776698 711714 360090 615052 394027 483319 670340 414898 070787 763612 561593 985604 > 4²²⁹ [i]