MinT - Best Known (168−57, 168, s)-Nets in Base 4

Best Known (168−57, 168, s)-Nets in Base 4

(168−57, 168, 160)-Net over F₄ — Constructive and digital

Digital (111, 168, 160)-net over F₄, using

4¹ times duplication [i] based on digital (110, 167, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 41, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (69, 126, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 63, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(168−57, 168, 208)-Net in Base 4 — Constructive

(111, 168, 208)-net in base 4, using

2 times m-reduction [i] based on (111, 170, 208)-net in base 4, using
- trace code for nets [i] based on (26, 85, 104)-net in base 16, using
  - base change [i] based on digital (9, 68, 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]

(168−57, 168, 444)-Net over F₄ — Digital

Digital (111, 168, 444)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−57, 168, 14657)-Net in Base 4 — Upper bound on s

There is no (111, 168, 14658)-net in base 4, because

1 times m-reduction [i] would yield (111, 167, 14658)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35015 701536 317602 228266 595367 794572 748517 481300 254908 485325 267364 620834 036750 230291 489902 003645 968720 > 4¹⁶⁷ [i]