MinT - Best Known (255−95, 255, s)-Nets in Base 4

Best Known (255−95, 255, s)-Nets in Base 4

(255−95, 255, 160)-Net over F₄ — Constructive and digital

Digital (160, 255, 160)-net over F₄, using

t-expansion [i] based on digital (157, 255, 160)-net over F₄, using
- 4 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 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, 175, 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]

(255−95, 255, 460)-Net over F₄ — Digital

Digital (160, 255, 460)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−95, 255, 10942)-Net in Base 4 — Upper bound on s

There is no (160, 255, 10943)-net in base 4, because

1 times m-reduction [i] would yield (160, 254, 10943)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 838 190629 967137 045534 601884 144567 340466 750229 298700 165178 379737 792191 997808 375656 585541 220676 886714 734232 202813 039456 065070 718849 371491 726429 030180 521420 > 4²⁵⁴ [i]