MinT - Best Known (123, 188, s)-Nets in Base 4

Best Known (123, 188, s)-Nets in Base 4

(123, 188, 160)-Net over F₄ — Constructive and digital

Digital (123, 188, 160)-net over F₄, using

4¹ times duplication [i] based on digital (122, 187, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 45, 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 (77, 142, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 71, 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]

(123, 188, 208)-Net in Base 4 — Constructive

(123, 188, 208)-net in base 4, using

2 times m-reduction [i] based on (123, 190, 208)-net in base 4, using
- trace code for nets [i] based on (28, 95, 104)-net in base 16, using
  - base change [i] based on digital (9, 76, 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]

(123, 188, 457)-Net over F₄ — Digital

Digital (123, 188, 457)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 188, 14035)-Net in Base 4 — Upper bound on s

There is no (123, 188, 14036)-net in base 4, because

1 times m-reduction [i] would yield (123, 187, 14036)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38491 217124 571841 496752 964172 913400 450466 443023 500800 731873 050924 475629 635180 830976 997294 777919 157161 037693 599350 > 4¹⁸⁷ [i]