MinT - Best Known (190−117, 190, s)-Nets in Base 4

Best Known (190−117, 190, s)-Nets in Base 4

(190−117, 190, 104)-Net over F₄ — Constructive and digital

Digital (73, 190, 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]

(190−117, 190, 112)-Net over F₄ — Digital

Digital (73, 190, 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]

(190−117, 190, 638)-Net in Base 4 — Upper bound on s

There is no (73, 190, 639)-net in base 4, because

1 times m-reduction [i] would yield (73, 189, 639)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 622411 437335 252175 975560 454307 102848 559832 819090 238376 932820 805307 013522 311976 036397 319908 486164 844163 883476 245335 > 4¹⁸⁹ [i]