MinT - Best Known (249−94, 249, s)-Nets in Base 4

Best Known (249−94, 249, s)-Nets in Base 4

(249−94, 249, 160)-Net over F₄ — Constructive and digital

Digital (155, 249, 160)-net over F₄, using

4 times m-reduction [i] based on digital (155, 253, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 82, 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, 171, 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]

(249−94, 249, 431)-Net over F₄ — Digital

Digital (155, 249, 431)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−94, 249, 9437)-Net in Base 4 — Upper bound on s

There is no (155, 249, 9438)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 821703 302296 738380 184905 285639 373761 899358 752062 986637 958972 893421 855063 307976 776296 263600 898528 202898 799435 037599 245378 762050 924890 367705 217989 092560 > 4²⁴⁹ [i]