MinT - Best Known (226−57, 226, s)-Nets in Base 4

Best Known (226−57, 226, s)-Nets in Base 4

(226−57, 226, 548)-Net over F₄ — Constructive and digital

Digital (169, 226, 548)-net over F₄, using

4¹ times duplication [i] based on 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]

(226−57, 226, 648)-Net in Base 4 — Constructive

(169, 226, 648)-net in base 4, using

t-expansion [i] based on (168, 226, 648)-net in base 4, using
- 2 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]

(226−57, 226, 1974)-Net over F₄ — Digital

Digital (169, 226, 1974)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−57, 226, 259314)-Net in Base 4 — Upper bound on s

There is no (169, 226, 259315)-net in base 4, because

1 times m-reduction [i] would yield (169, 225, 259315)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2907 360342 038041 391893 669893 733369 101260 501927 642301 515483 936632 440894 895881 828187 291851 234464 310260 300700 070808 927664 604945 316274 342280 > 4²²⁵ [i]