MinT - Best Known (63, 63+133, s)-Nets in Base 4

Best Known (63, 63+133, s)-Nets in Base 4

(63, 63+133, 66)-Net over F₄ — Constructive and digital

Digital (63, 196, 66)-net over F₄, using

t-expansion [i] based on digital (49, 196, 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+133, 99)-Net over F₄ — Digital

Digital (63, 196, 99)-net over F₄, using

t-expansion [i] based on digital (61, 196, 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+133, 456)-Net in Base 4 — Upper bound on s

There is no (63, 196, 457)-net in base 4, because

1 times m-reduction [i] would yield (63, 195, 457)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2627 256439 897820 829896 075357 556877 240158 538693 981877 581579 784242 214839 096142 432772 016761 598616 209956 383353 950959 864740 > 4¹⁹⁵ [i]