MinT - Best Known (166−93, 166, s)-Nets in Base 4

Best Known (166−93, 166, s)-Nets in Base 4

(166−93, 166, 104)-Net over F₄ — Constructive and digital

Digital (73, 166, 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]

(166−93, 166, 112)-Net over F₄ — Digital

Digital (73, 166, 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]

(166−93, 166, 829)-Net in Base 4 — Upper bound on s

There is no (73, 166, 830)-net in base 4, because

1 times m-reduction [i] would yield (73, 165, 830)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2295 380261 402043 144348 711329 951463 721451 453978 767149 615268 542142 785749 218266 557015 913198 166587 698128 > 4¹⁶⁵ [i]