MinT - Best Known (252−97, 252, s)-Nets in Base 4

Best Known (252−97, 252, s)-Nets in Base 4

(252−97, 252, 160)-Net over F₄ — Constructive and digital

Digital (155, 252, 160)-net over F₄, using

1 times m-reduction [i] based on digital (155, 253, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 82, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 171, 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]

(252−97, 252, 407)-Net over F₄ — Digital

Digital (155, 252, 407)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−97, 252, 8749)-Net in Base 4 — Upper bound on s

There is no (155, 252, 8750)-net in base 4, because

1 times m-reduction [i] would yield (155, 251, 8750)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 122930 876617 204535 906527 554047 703290 935585 840965 324191 825153 799717 573279 820459 024092 875583 425908 502431 693966 174463 932017 078096 672876 580168 933768 366176 > 4²⁵¹ [i]