MinT - Best Known (192−73, 192, s)-Nets in Base 4

Best Known (192−73, 192, s)-Nets in Base 4

(192−73, 192, 131)-Net over F₄ — Constructive and digital

Digital (119, 192, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 46, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 146, 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]

(192−73, 192, 335)-Net over F₄ — Digital

Digital (119, 192, 335)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(192−73, 192, 7415)-Net in Base 4 — Upper bound on s

There is no (119, 192, 7416)-net in base 4, because

1 times m-reduction [i] would yield (119, 191, 7416)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 860871 040389 992777 601558 392941 640538 650215 205091 543047 498395 667331 150722 657381 554172 115338 850177 668909 198470 626475 > 4¹⁹¹ [i]