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

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

(168, 168+57, 548)-Net over F₄ — Constructive and digital

Digital (168, 225, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 33, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(168, 168+57, 648)-Net in Base 4 — Constructive

(168, 225, 648)-net in base 4, using

3 times m-reduction [i] based on (168, 228, 648)-net in base 4, using
- trace code for nets [i] based on (16, 76, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(168, 168+57, 1927)-Net over F₄ — Digital

Digital (168, 225, 1927)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+57, 246787)-Net in Base 4 — Upper bound on s

There is no (168, 225, 246788)-net in base 4, because

1 times m-reduction [i] would yield (168, 224, 246788)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 862702 538758 706984 872161 248465 226654 043687 183398 153721 213714 891026 919276 172627 589377 231053 963409 204555 946781 305059 623879 452184 472320 > 4²²⁴ [i]