MinT - Best Known (242−101, 242, s)-Nets in Base 4

Best Known (242−101, 242, s)-Nets in Base 4

(242−101, 242, 137)-Net over F₄ — Constructive and digital

Digital (141, 242, 137)-net over F₄, using

5 times m-reduction [i] based on digital (141, 247, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 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, 179, 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]

(242−101, 242, 302)-Net over F₄ — Digital

Digital (141, 242, 302)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−101, 242, 5140)-Net in Base 4 — Upper bound on s

There is no (141, 242, 5141)-net in base 4, because

1 times m-reduction [i] would yield (141, 241, 5141)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 532406 866749 906214 255560 986621 391717 258842 791195 595543 524100 975458 047948 118906 751488 993018 095973 419744 984826 597267 100075 028871 661664 931373 039296 > 4²⁴¹ [i]