MinT - Best Known (234−84, 234, s)-Nets in Base 4

Best Known (234−84, 234, s)-Nets in Base 4

(234−84, 234, 160)-Net over F₄ — Constructive and digital

Digital (150, 234, 160)-net over F₄, using

4 times m-reduction [i] based on digital (150, 238, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 77, 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, 161, 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]

(234−84, 234, 208)-Net in Base 4 — Constructive

(150, 234, 208)-net in base 4, using

trace code for nets [i] based on (33, 117, 104)-net in base 16, using
- 3 times m-reduction [i] based on (33, 120, 104)-net in base 16, using
  - base change [i] based on digital (9, 96, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(234−84, 234, 486)-Net over F₄ — Digital

Digital (150, 234, 486)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−84, 234, 12411)-Net in Base 4 — Upper bound on s

There is no (150, 234, 12412)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 764 442953 534530 746741 997421 753964 521329 111121 366540 820807 691154 733749 615133 852935 629637 373722 412006 394955 118399 340303 914510 919844 266748 545285 > 4²³⁴ [i]