MinT - Best Known (212−86, 212, s)-Nets in Base 4

Best Known (212−86, 212, s)-Nets in Base 4

(212−86, 212, 131)-Net over F₄ — Constructive and digital

Digital (126, 212, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 53, 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, 159, 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]

(212−86, 212, 294)-Net over F₄ — Digital

Digital (126, 212, 294)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−86, 212, 5196)-Net in Base 4 — Upper bound on s

There is no (126, 212, 5197)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 515555 199145 155186 231670 093553 083901 692061 072947 165540 400385 925910 246831 154425 418778 115371 736662 831607 989010 391841 021434 506760 > 4²¹² [i]