MinT - Best Known (225−85, 225, s)-Nets in Base 4

Best Known (225−85, 225, s)-Nets in Base 4

(225−85, 225, 139)-Net over F₄ — Constructive and digital

Digital (140, 225, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 43, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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]

(225−85, 225, 393)-Net over F₄ — Digital

Digital (140, 225, 393)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−85, 225, 8912)-Net in Base 4 — Upper bound on s

There is no (140, 225, 8913)-net in base 4, because

1 times m-reduction [i] would yield (140, 224, 8913)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 729 092231 515062 635749 226771 822359 526215 315659 915025 259643 836575 939185 671215 916205 197414 705337 657100 429722 528130 123813 324130 824585 825360 > 4²²⁴ [i]