MinT - Best Known (214−81, 214, s)-Nets in Base 4

Best Known (214−81, 214, s)-Nets in Base 4

(214−81, 214, 137)-Net over F₄ — Constructive and digital

Digital (133, 214, 137)-net over F₄, using

9 times m-reduction [i] based on digital (133, 223, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 60, 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, 163, 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]

(214−81, 214, 373)-Net over F₄ — Digital

Digital (133, 214, 373)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−81, 214, 8412)-Net in Base 4 — Upper bound on s

There is no (133, 214, 8413)-net in base 4, because

1 times m-reduction [i] would yield (133, 213, 8413)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 302552 286490 805060 880107 966042 262773 305079 945340 004100 043687 354584 125058 424867 708897 781008 849510 154596 631573 676137 790634 233870 > 4²¹³ [i]