MinT - Best Known (237−86, 237, s)-Nets in Base 4

Best Known (237−86, 237, s)-Nets in Base 4

(237−86, 237, 160)-Net over F₄ — Constructive and digital

Digital (151, 237, 160)-net over F₄, using

4 times m-reduction [i] based on digital (151, 241, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 78, 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, 163, 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]

(237−86, 237, 196)-Net in Base 4 — Constructive

(151, 237, 196)-net in base 4, using

3 times m-reduction [i] based on (151, 240, 196)-net in base 4, using
- trace code for nets [i] based on (31, 120, 98)-net in base 16, using
  - base change [i] based on digital (7, 96, 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]

(237−86, 237, 473)-Net over F₄ — Digital

Digital (151, 237, 473)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−86, 237, 11677)-Net in Base 4 — Upper bound on s

There is no (151, 237, 11678)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48782 481257 913747 012568 563860 548594 260514 169197 959687 796122 611398 807340 726466 643040 047147 034550 329817 505300 191865 659683 786143 495795 671515 734600 > 4²³⁷ [i]