MinT - Best Known (251−104, 251, s)-Nets in Base 4

Best Known (251−104, 251, s)-Nets in Base 4

(251−104, 251, 138)-Net over F₄ — Constructive and digital

Digital (147, 251, 138)-net over F₄, using

2 times m-reduction [i] based on digital (147, 253, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 179, 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]

(251−104, 251, 320)-Net over F₄ — Digital

Digital (147, 251, 320)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−104, 251, 5388)-Net in Base 4 — Upper bound on s

There is no (147, 251, 5389)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 192785 543305 902980 656464 962408 283223 402809 411127 447420 479423 880029 888203 254441 952076 978711 820616 261937 841447 996079 001128 486697 905149 851190 560411 443280 > 4²⁵¹ [i]