MinT - Best Known (172−95, 172, s)-Nets in Base 4

Best Known (172−95, 172, s)-Nets in Base 4

(172−95, 172, 104)-Net over F₄ — Constructive and digital

Digital (77, 172, 104)-net over F₄, using

t-expansion [i] based on digital (73, 172, 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]

(172−95, 172, 112)-Net over F₄ — Digital

Digital (77, 172, 112)-net over F₄, using

t-expansion [i] based on digital (73, 172, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(172−95, 172, 911)-Net in Base 4 — Upper bound on s

There is no (77, 172, 912)-net in base 4, because

1 times m-reduction [i] would yield (77, 171, 912)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 261774 703989 131969 860289 443997 180023 459942 336474 384143 263651 323675 165914 094057 865173 034949 412009 356940 > 4¹⁷¹ [i]