MinT - Best Known (242−102, 242, s)-Nets in Base 4

Best Known (242−102, 242, s)-Nets in Base 4

(242−102, 242, 137)-Net over F₄ — Constructive and digital

Digital (140, 242, 137)-net over F₄, using

2 times m-reduction [i] based on digital (140, 244, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 67, 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, 177, 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]

(242−102, 242, 292)-Net over F₄ — Digital

Digital (140, 242, 292)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−102, 242, 4717)-Net in Base 4 — Upper bound on s

There is no (140, 242, 4718)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 099892 550582 994219 768059 995541 543954 496715 927701 245039 790676 941222 517738 783561 308836 527082 246842 615506 730990 369466 941015 769726 984637 694064 525000 > 4²⁴² [i]