MinT - Best Known (137, 137+84, s)-Nets in Base 4

Best Known (137, 137+84, s)-Nets in Base 4

(137, 137+84, 138)-Net over F₄ — Constructive and digital

Digital (137, 221, 138)-net over F₄, using

2 times m-reduction [i] based on digital (137, 223, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 64, 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, 159, 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]

(137, 137+84, 379)-Net over F₄ — Digital

Digital (137, 221, 379)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+84, 8068)-Net in Base 4 — Upper bound on s

There is no (137, 221, 8069)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 364198 327460 047394 064276 949556 617304 544373 305221 702032 225165 916858 819749 226916 406277 288717 694966 384976 741833 691919 059059 890050 383200 > 4²²¹ [i]