MinT - Best Known (249−95, 249, s)-Nets in Base 4

Best Known (249−95, 249, s)-Nets in Base 4

(249−95, 249, 160)-Net over F₄ — Constructive and digital

Digital (154, 249, 160)-net over F₄, using

1 times m-reduction [i] based on digital (154, 250, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 81, 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, 169, 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]

(249−95, 249, 415)-Net over F₄ — Digital

Digital (154, 249, 415)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−95, 249, 9161)-Net in Base 4 — Upper bound on s

There is no (154, 249, 9162)-net in base 4, because

1 times m-reduction [i] would yield (154, 248, 9162)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204833 342843 241984 565539 653594 763788 104080 300984 144866 144850 943678 587634 760789 222083 890163 744462 278016 272755 731351 637359 925254 266933 612153 022971 046240 > 4²⁴⁸ [i]