MinT - Best Known (63, 63+137, s)-Nets in Base 4

Best Known (63, 63+137, s)-Nets in Base 4

(63, 63+137, 66)-Net over F₄ — Constructive and digital

Digital (63, 200, 66)-net over F₄, using

t-expansion [i] based on digital (49, 200, 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]

(63, 63+137, 99)-Net over F₄ — Digital

Digital (63, 200, 99)-net over F₄, using

t-expansion [i] based on digital (61, 200, 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]

(63, 63+137, 450)-Net in Base 4 — Upper bound on s

There is no (63, 200, 451)-net in base 4, because

1 times m-reduction [i] would yield (63, 199, 451)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 738642 966319 248483 466635 985028 176373 591359 656006 087009 947273 443535 247785 479156 019073 380481 650901 023202 585802 788791 377600 > 4¹⁹⁹ [i]