MinT - Best Known (212−82, 212, s)-Nets in Base 4

Best Known (212−82, 212, s)-Nets in Base 4

(212−82, 212, 137)-Net over F₄ — Constructive and digital

Digital (130, 212, 137)-net over F₄, using

2 times m-reduction [i] based on digital (130, 214, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 57, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 157, 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]

(212−82, 212, 344)-Net over F₄ — Digital

Digital (130, 212, 344)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−82, 212, 6946)-Net in Base 4 — Upper bound on s

There is no (130, 212, 6947)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 366772 508792 009531 107684 540590 626570 439414 861279 112283 278212 844711 672206 773518 336767 538469 584511 971311 367177 265252 671145 367360 > 4²¹² [i]