MinT - Best Known (219−85, 219, s)-Nets in Base 4

Best Known (219−85, 219, s)-Nets in Base 4

(219−85, 219, 137)-Net over F₄ — Constructive and digital

Digital (134, 219, 137)-net over F₄, using

7 times m-reduction [i] based on digital (134, 226, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 61, 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, 165, 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]

(219−85, 219, 350)-Net over F₄ — Digital

Digital (134, 219, 350)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−85, 219, 7304)-Net in Base 4 — Upper bound on s

There is no (134, 219, 7305)-net in base 4, because

1 times m-reduction [i] would yield (134, 218, 7305)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177522 074773 551043 806068 252551 436125 021499 031104 240932 829750 288081 711180 658126 013040 375117 541359 693761 162593 410849 304978 468577 006720 > 4²¹⁸ [i]