MinT - Best Known (130, 130+89, s)-Nets in Base 4

Best Known (130, 130+89, s)-Nets in Base 4

(130, 130+89, 134)-Net over F₄ — Constructive and digital

Digital (130, 219, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 57, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 162, 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]

(130, 130+89, 301)-Net over F₄ — Digital

Digital (130, 219, 301)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 130+89, 5494)-Net in Base 4 — Upper bound on s

There is no (130, 219, 5495)-net in base 4, because

1 times m-reduction [i] would yield (130, 218, 5495)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 178797 220168 722593 534473 852622 104628 233668 866394 285350 044198 751262 216033 283797 140413 281696 461819 826387 103497 358414 553997 010808 706982 > 4²¹⁸ [i]