MinT - Best Known (212−80, 212, s)-Nets in Base 4

Best Known (212−80, 212, s)-Nets in Base 4

(212−80, 212, 137)-Net over F₄ — Constructive and digital

Digital (132, 212, 137)-net over F₄, using

8 times m-reduction [i] based on digital (132, 220, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 59, 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, 161, 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]

(212−80, 212, 374)-Net over F₄ — Digital

Digital (132, 212, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−80, 212, 8125)-Net in Base 4 — Upper bound on s

There is no (132, 212, 8126)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 473034 818987 498588 227969 016303 738029 288481 895897 030241 926723 717636 134549 536162 428040 854293 629251 647847 809440 208369 541637 559451 > 4²¹² [i]