MinT - Best Known (251−95, 251, s)-Nets in Base 4

Best Known (251−95, 251, s)-Nets in Base 4

(251−95, 251, 160)-Net over F₄ — Constructive and digital

Digital (156, 251, 160)-net over F₄, using

5 times m-reduction [i] based on digital (156, 256, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 83, 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, 173, 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]

(251−95, 251, 430)-Net over F₄ — Digital

Digital (156, 251, 430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−95, 251, 9720)-Net in Base 4 — Upper bound on s

There is no (156, 251, 9721)-net in base 4, because

1 times m-reduction [i] would yield (156, 250, 9721)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 276070 985440 221594 349253 263781 046288 320317 892260 719837 916948 085632 627906 157987 794376 612831 881013 248634 140807 771171 522375 246023 790023 936249 624430 211248 > 4²⁵⁰ [i]