MinT - Best Known (63, 63+158, s)-Nets in Base 4

Best Known (63, 63+158, s)-Nets in Base 4

Digital (63, 221, 66)-net over F₄, using

t-expansion [i] based on digital (49, 221, 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, 221, 99)-net over F₄, using

t-expansion [i] based on digital (61, 221, 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, 221, 390)-net over F₄, because

2 times m-reduction [i] would yield digital (63, 219, 390)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁹, 390, F₄, 156) (dual of [390, 171, 157]-code), but
  - residual code [i] would yield OA(4⁶³, 233, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 2 394144 335707 471520 884224 011051 775831 492539 148504 246999 784192 681791 545828 966400 000000 / 26753 789254 137656 696530 296222 884590 520869 895903 > 4⁶³ [i]

There is no (63, 221, 425)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 759224 423103 657942 799561 477884 579335 856662 867681 726896 592524 699674 501665 779227 439486 116951 237672 799723 622021 591434 063041 212886 671168 > 4²²¹ [i]