MinT - Best Known (158−77, 158, s)-Nets in Base 4

Best Known (158−77, 158, s)-Nets in Base 4

(158−77, 158, 104)-Net over F₄ — Constructive and digital

Digital (81, 158, 104)-net over F₄, using

t-expansion [i] based on digital (73, 158, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(158−77, 158, 129)-Net over F₄ — Digital

Digital (81, 158, 129)-net over F₄, using

net from sequence [i] based on digital (81, 128)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
  - Drinfeld modules of rank 1 [i]

(158−77, 158, 1507)-Net in Base 4 — Upper bound on s

There is no (81, 158, 1508)-net in base 4, because

1 times m-reduction [i] would yield (81, 157, 1508)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33602 658915 913992 890240 237696 119596 240553 655531 788568 295062 148957 117940 767544 860909 598824 143920 > 4¹⁵⁷ [i]