MinT - Best Known (248−92, 248, s)-Nets in Base 4

Best Known (248−92, 248, s)-Nets in Base 4

(248−92, 248, 160)-Net over F₄ — Constructive and digital

Digital (156, 248, 160)-net over F₄, using

8 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]

(248−92, 248, 196)-Net in Base 4 — Constructive

(156, 248, 196)-net in base 4, using

trace code for nets [i] based on (32, 124, 98)-net in base 16, using
- 1 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
  - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(248−92, 248, 456)-Net over F₄ — Digital

Digital (156, 248, 456)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−92, 248, 10530)-Net in Base 4 — Upper bound on s

There is no (156, 248, 10531)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 205189 176521 699377 066181 549711 279831 676487 393966 201551 677024 477875 098104 595806 121120 194844 426839 153547 690118 621228 372915 013997 721685 941212 575873 973952 > 4²⁴⁸ [i]