MinT - Best Known (247−105, 247, s)-Nets in Base 4

Best Known (247−105, 247, s)-Nets in Base 4

(247−105, 247, 137)-Net over F₄ — Constructive and digital

Digital (142, 247, 137)-net over F₄, using

3 times m-reduction [i] based on digital (142, 250, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 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, 181, 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]

(247−105, 247, 290)-Net over F₄ — Digital

Digital (142, 247, 290)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−105, 247, 4710)-Net in Base 4 — Upper bound on s

There is no (142, 247, 4711)-net in base 4, because

1 times m-reduction [i] would yield (142, 246, 4711)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12860 096862 217290 793448 682699 996288 099812 600312 087461 775662 589842 818165 859316 824062 427150 856038 783744 906119 172324 656565 878854 686262 483593 868749 749940 > 4²⁴⁶ [i]